Exploring the Riemann Zeta function : 190 years from Riemann's birth 978-3-319-59969-4, 3319599690, 978-3-319-59968-7

Table of contents :
Front Matter ....Pages i-x
An Introduction to Riemann’s Life, His Mathematics, and His Work on the Zeta Function (Roger Baker)....Pages 1-12
Ramanujan’s Formula for ζ(2n + 1) (Bruce C. Berndt, Armin Straub)....Pages 13-34
Towards a Fractal Cohomology: Spectra of Polya–Hilbert Operators, Regularized Determinants and Riemann Zeros (Tim Cobler, Michel L. Lapidus)....Pages 35-65
The Temptation of the Exceptional Characters (John B. Friedlander, Henryk Iwaniec)....Pages 67-81
Arthur’s Truncated Eisenstein Series for SL(2, Z) and the Riemann Zeta Function: A Survey (Dorian Goldfeld)....Pages 83-97
On a Cubic Moment of Hardy’s Function with a Shift (Aleksandar Ivić)....Pages 99-112
Some Analogues of Pair Correlation of Zeta Zeros (Yunus Karabulut, Cem Yalçın Yıldırım)....Pages 113-179
Bagchi’s Theorem for Families of Automorphic Forms (E. Kowalski)....Pages 181-199
The Liouville Function and the Riemann Hypothesis (Michael J. Mossinghoff, Timothy S. Trudgian)....Pages 201-221
Explorations in the Theory of Partition Zeta Functions (Ken Ono, Larry Rolen, Robert Schneider)....Pages 223-264
Reading Riemann (S. J. Patterson)....Pages 265-285
A Taniyama Product for the Riemann Zeta Function (David E. Rohrlich)....Pages 287-298

Citation preview

Hugh Montgomery · Ashkan Nikeghbali Michael Th. Rassias Editors

Exploring the Riemann Zeta Function 190 years from Riemann’s Birth With a preface by Freeman J. Dyson

Exploring the Riemann Zeta Function

Hugh Montgomery • Ashkan Nikeghbali Michael Th. Rassias Editors

Exploring the Riemann Zeta Function 190 years from Riemann’s Birth

With a preface by Freeman J. Dyson

123

Editors Hugh Montgomery Department of Mathematics University of Michigan Ann Arbor, MI, USA

Ashkan Nikeghbali Institut für Mathematik Universität Zürich Zürich, Switzerland

Michael Th. Rassias Institut für Mathematik Universität Zürich Zürich, Switzerland

ISBN 978-3-319-59968-7 ISBN 978-3-319-59969-4 (eBook) DOI 10.1007/978-3-319-59969-4 Library of Congress Control Number: 2017947901 Mathematics Subject Classification (2010): 11-XX, 30-XX, 33-XX, 41-XX, 43-XX, 60-XX © Springer International Publishing AG 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface by Freeman J. Dyson: Quasi-Crystals and the Riemann Hypothesis

On the occasion of the 190 years from Riemann’s birth, which led to the conception of this book, I am presenting a challenge to young mathematicians to study quasicrystals and use them to prove the Riemann Hypothesis. Among the readers, those who are experts in number theory may consider the challenge frivolous. Those who are not experts may consider it uninteresting or unintelligible. Nevertheless, I am putting it forward for your serious consideration. One should always be guided by the wisdom of the physicist Leo Szilard, who wrote his own Ten Commandments to stand beside those of Moses. Szilard’s second commandment says, “Let your acts be directed toward a worthy goal, but do not ask if they will reach it; they are to be models and examples, not means to an end.” The Riemann Hypothesis is a worthy goal, and it is not for us to ask whether we can reach it. I will give you some hints describing how it might be achieved. Here, I will be giving voice to the mathematician that I was almost 70 years ago before I became a physicist. There were until recently two supreme unsolved problems in the world of pure mathematics, the proof of Fermat’s Last Theorem and the proof of the Riemann Hypothesis. In 1998, my former Princeton colleague Andrew Wiles polished off Fermat’s Last Theorem, and only the Riemann Hypothesis remains. Wiles’ proof of the Fermat Theorem was not just a technical stunt. It required the discovery and exploration of a new field of mathematical ideas, far wider and more consequential than the Fermat Theorem itself. It is likely that any proof of the Riemann Hypothesis will likewise lead to a deeper understanding of many diverse areas of mathematics and perhaps of physics too. The Riemann Hypothesis says that one specific function, the zeta-function that Riemann named, has all its complex zeros upon a certain line. Riemann’s zeta-function and other zeta-functions similar to it appear ubiquitously in number theory, in the theory of dynamical systems, in geometry, in function theory, and in physics. The zeta-function stands at a junction where paths lead in many directions. A proof of the hypothesis will illuminate all the connections. Like every serious student of pure mathematics, when I was young, I had dreams of proving the Riemann Hypothesis. I had some vague ideas that I thought might lead to a proof, but never pursued them vigorously. In recent years, after the discovery

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Preface by Freeman J. Dyson: Quasi-Crystals and the Riemann Hypothesis

of quasicrystals, my ideas became a little less vague. I offer them here for the consideration of any young mathematician who has ambitions to win a Fields Medal. Now, I jump from the Riemann Hypothesis to quasicrystals. Quasicrystals were one of the great unexpected discoveries of recent years. They were discovered in two ways, as real physical objects formed when alloys of aluminum and manganese are solidified rapidly from the molten state and as abstract mathematical structures in Euclidean geometry. They were unexpected, both in physics and in mathematics. I am here discussing only their mathematical properties. Quasicrystals can exist in spaces of one, two, or three dimensions. From the point of view of physics, the three-dimensional quasicrystals are the most interesting, since they inhabit our three-dimensional world and can be studied experimentally. From the point of view of a mathematician, one-dimensional quasicrystals are more interesting than two-dimensional quasicrystals because they exist in far greater variety, and the two-dimensional are more interesting than the three-dimensional for the same reason. The mathematical definition of a quasicrystal is as follows. A quasicrystal is a distribution of discrete point masses whose Fourier transform is a distribution of discrete point frequencies. Or to say it more briefly, a quasicrystal is a pure point distribution that has a pure point spectrum. This definition includes as a special case the ordinary crystals which are exactly periodic point distributions with exactly periodic point spectra. Excluding the ordinary crystals, quasicrystals in three dimensions come in very limited variety, all of them being associated with the icosahedral rotation group. The two-dimensional quasicrystals are more numerous, roughly one distinct type associated with each regular polygon in a plane. The two-dimensional quasicrystal with pentagonal symmetry is the famous Penrose tiling of the plane, discovered by Penrose before the general concept of quasicrystal was invented. Finally, the one-dimensional quasicrystals have a far richer structure since they are not tied to any rotational symmetries. So far as I know, no complete enumeration of onedimensional quasicrystals exists. It is known that a unique quasi-crystal exists corresponding to every Pisot-Vijayaraghavan number or PV number. A PV number is a real algebraic integer, a root of a polynomial equation with integer coefficients, such that all the other roots have absolute value less than one (see [1]). The set of all PV numbers is infinite and has a remarkable topological structure. The set of all one-dimensional quasicrystals has a structure at least as rich as the set of all PV numbers and probably much richer. We do not know for sure, but it is likely that a huge universe of one-dimensional quasicrystals not associated with PV numbers is waiting to be discovered. Here comes the punch line of my argument, the essential point linking onedimensional quasicrystals with the Riemann Hypothesis. If the Riemann Hypothesis is true, then the zeros of the zeta-function form a one-dimensional quasicrystal according to the definition. They constitute a distribution of point masses on a straight line, and their Fourier transform is likewise a distribution of point masses, one at each of the logarithms of ordinary prime-power numbers. Andrew Odlyzko (see [2]) has published a beautiful computer calculation of the Fourier transform

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of the zeta-function zeros. The calculation shows precisely the expected structure of the Fourier transform, with a sharp discontinuity at every logarithm of a primepower number and nowhere else. My proposal is the following. Let us pretend that we do not know that the Riemann Hypothesis is true. Let us tackle the problem from the other end. Let us try to obtain a complete enumeration and classification of all one-dimensional quasicrystals. That is to say, we enumerate and classify all point distributions that have a discrete point spectrum. We shall then find the well-known quasicrystals associated with PV numbers and also a whole slew of other quasicrystals, known and unknown. Among the slew of other quasicrystals, we should be able to identify one corresponding to the Riemann zeta-function and one corresponding to each of the other zeta-functions that resemble the Riemann zeta-function. Suppose that we can prove rigorously that one of the quasicrystals in our enumeration has properties that identify it with the zeros of the Riemann zeta-function. Then we have proved the Riemann Hypothesis and can wait for the telephone call announcing the award of the Fields Medal. These are of course idle dreams. The problem of classifying one-dimensional quasicrystals is horrendously difficult, probably at least as difficult as the problems that Andrew Wiles took 7 years to fight his way through. But still, the history of mathematics is a history of horrendously difficult problems being solved by young people too ignorant to know that they were impossible. The classification of quasicrystals is a worthy goal and might even turn out to be achievable. Problems of that degree of difficulty will not be solved by old men like me. Let the young people now have a try. Institute for Advanced Study 1 Einstein Dr Princeton, NJ 08540, USA

Freeman J. Dyson

References 1. M.J. Bertin et al., Pisot and Salem Numbers (Birkhäuser Verlag, Basel, 1992) 2. A.M. Odlyzko, Primes, quantum chaos and computers, in Number Theory, Proceedings of a Symposium (National Research Council, Washington, DC, 1990), pp. 35–46

Contents

Preface by Freeman J. Dyson: Quasi-Crystals and the Riemann Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An Introduction to Riemann’s Life, His Mathematics, and His Work on the Zeta Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Roger Baker Ramanujan’s Formula for (2n C 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bruce C. Berndt and Armin Straub Towards a Fractal Cohomology: Spectra of Polya–Hilbert Operators, Regularized Determinants and Riemann Zeros . . . . . . . . . . . . . . . . . Tim Cobler and Michel L. Lapidus The Temptation of the Exceptional Characters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . John B. Friedlander and Henryk Iwaniec Arthur’s Truncated Eisenstein Series for SL(2, Z) and the Riemann Zeta Function: A Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dorian Goldfeld On a Cubic Moment of Hardy’s Function with a Shift . . . . . . . . . . . . . . . . . . . . . . Aleksandar Ivi´c

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35 67

83 99

Some Analogues of Pair Correlation of Zeta Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Yunus Karabulut and Cem Yalçın Yıldırım Bagchi’s Theorem for Families of Automorphic Forms. . . . . . . . . . . . . . . . . . . . . . 181 E. Kowalski The Liouville Function and the Riemann Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . 201 Michael J. Mossinghoff and Timothy S. Trudgian Explorations in the Theory of Partition Zeta Functions . . . . . . . . . . . . . . . . . . . . . 223 Ken Ono, Larry Rolen, and Robert Schneider

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Reading Riemann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 S.J. Patterson A Taniyama Product for the Riemann Zeta Function . . . . . . . . . . . . . . . . . . . . . . . 287 David E. Rohrlich

An Introduction to Riemann’s Life, His Mathematics, and His Work on the Zeta Function Roger Baker

Abstract Although the zeta function was first defined and used by Euler, it is to Bernhard Riemann, in an article written in 1859, that we owe our view of the zeta function as a meromorphic function in the plane with a functional equation. Riemann is a very remarkable figure in the history of mathematics. The present article describes his career including the major mathematical highlights, and gives some discussion of his published and unpublished work on the zeta function.

1 Introduction In 1846 a frail and painfully shy youth of nineteen arrived in Göttingen, in the Kingdom of Hanover, to begin his university studies. Bernhard Riemann was to spend most of his adult life in Göttingen as an undergraduate, doctoral candidate, Privatdozent (junior university instructor), extraordinary professor (roughly, associate professor), and full professor. The best of his research papers, written in the 1850s, were among the very most influential papers in the mathematics of his century. In particular, in a paper written in 1859, he set Euler’s zeta function (as he would have referred to it) on a modern footing. Today, zeta functions are perhaps the part of number theory that we would most like to understand better, as indeed the publication of the present volume testifies. Georg Friedrich Bernhard Riemann was born on 17 September 1826 in Breselenz, in the Kingdom of Hanover. His father Friedrich was the local pastor; later he took over the parish of Quickborn, not far from the Free Imperial City of Hamburg, but still in the Kingdom of Hanover. Bernhard was tutored by his father up to the age of thirteen, when he moved to the city of Hanover; he lived with his grandmother and attended high school for 2 years. After her death, he attended the high school in Lüneburg, which is south-east of Hamburg. He was not notably successful except in mathematics. (At this time he read Legendre’s Théorie des Nombres [12].)

R. Baker () Department of Mathematics, Brigham Young University, Provo, UT 84602, USA e-mail: [email protected] © Springer International Publishing AG 2017 H. Montgomery et al. (eds.), Exploring the Riemann Zeta Function, DOI 10.1007/978-3-319-59969-4_1

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On arriving in Göttingen, Riemann initially matriculated as a student of philosophy and theology, perhaps at the time thinking of following his father’s profession. Certainly Riemann was very devout. But he soon focused on mathematics. In the summer of 1846, Riemann took Moritz Stern’s course on numerical solution of equations and Carl Goldschmidt’s course on terrestrial magnetism. Stern (1807–1894) and Goldschmidt (1807–1851) were both Gauss’s students. Stern was the first Jewish full professor in Germany and is known for the Stern–Brocot tree, with vertices corresponding to the positive rationals. Goldschmidt was Gauss’s assistant at the University Observatory and also helped Gauss gather data comparing the prime number counting function .x/ D

X

1

px

to the logarithmic integral Z li x D 0

x

dt log t

(this is a Cauchy principal value). Carl Friedrich Gauss (1777–1855) was deservedly famous for many contributions to mathematics, geodesy, and physics. By this time the elderly Gauss, who had been at Göttingen since 1807, would only give classes on his method of least squares, which included introductory linear algebra. Riemann took this course in winter 1846–1847, along with Stern’s course on definite integrals. Presumably in search of deeper mathematical content, Riemann spent the next 2 years, 1847–1849, at the University of Berlin. He attended Dirichlet’s courses in number theory, the theory of definite integrals, and partial differential equations; Jacobi’s lectures in analytical mechanics and linear algebra; and Eisenstein’s lectures on the theory of elliptic functions. We pause here to discuss the influence of these three fine mathematicians on Riemann. F.G.M. Eisenstein (1823–1852) worked mostly in number theory. Riemann recounted to Dedekind that they had discussed the use of complex numbers in the theory of functions, which seemed important to Riemann, and their views differed so radically that there was not much further interaction. C.G.J. Jacobi (1804–1851) was famous for his contributions to the theory of elliptic functions, which he based on four theta functions. He also worked on number theory and partial differential equations and dynamics. His collected works fill eight large volumes. Riemann was certainly very interested in the work on elliptic functions, as we shall see. However, the key person for his evolution as a mathematician was P.G.L. Dirichlet (1805–1859). Dirichlet was a brilliant lecturer and some of his lectures on number theory are still extant [5]. He was regarded by Jacobi as the only fully rigorous analyst. His paper of 1829 on pointwise convergence of Fourier series was a model of precision; this would be the area of Riemann’s habilitation thesis of 1854. His introduction of “Dirichlet characters” and the associated L functions

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[treated as functions on .0; 1/] were needed for his proof that there are infinitely many primes in any arithmetic progression a .mod q/ with .a; q/ D 1; this was completed in a paper of 1839–1840. Dirichlet’s wife Rebecka was the sister of the great Felix Mendelssohn-Bartholdy. We mention briefly the political events of 1848 and beyond. At this time Germany was a loosely associated group of 38 states including Hanover; the most important were Prussia, with Berlin as the capital, and Austria, with Vienna as the capital. Many of these states and other parts of Europe were shaken by revolution in 1848–1849. Ultimately, the revolution in German states was put down by military force, but modernization of political systems followed gradually. In 1862, Otto von Bismarck became Prussian prime minister. His skills, and Prussian military prowess, led by 1871 to the unification of Germany as an empire excluding Austria– Hungary. Bismarck was the Empire’s Chancellor from 1871 to 1890. Many of these events occurred after Riemann’s death, but as we shall see, he was affected by the war of 1866. In this war, Prussia (which had proposed a radical and unwelcome reshaping of the German confederation) faced Austria, which allied with most of the “third Germany” (states other than Prussia and Austria). Prussia won the battle of Königgratz (July 1866) and then annexed much north German territory, including the Kingdom of Hanover. Returning to March 1848, Riemann was loyal to the establishment and, as a member of the student corps, was on guard duty at the Royal Palace from 9 a.m. on 24 March to 1 p.m. the next day. This seems to have been the only time he took sides in political matters. At Easter 1849 Riemann returned to Göttingen and attended courses of lectures on philosophy and experimental physics. Here his teacher, Wilhelm Weber (1804–1891) was a remarkable man who also had much influence on Riemann. Weber became a professor of physics in Göttingen in 1831 at the instigation of Gauss. Gauss and Weber published important joint work on magnetism. Their famous telegraph, the first effective working one, dated from 1833 and permitted simultaneous magnetic observations at the Physical Laboratory and the Astronomical Observatory two miles away. There was an interruption in Weber’s work in 1837 when, with six other professors, he was dismissed for a protest concerning the revocation of the Hanoverian constitution of 1833. After traveling, and working at Leipzig, he was reinstated in 1849. His later work focused on electrodynamics. In 1850 a seminar in mathematical physics was formed, led by Weber, in which Riemann joined. To us it is remarkable that he could be heavily involved in experimental work at the same time he was preparing his groundbreaking doctoral thesis, “Foundations for a general theory of functions of a complex variable.” Riemann published papers in physics throughout his short life. His paper [23] seems to be the first in which it is asserted that electromagnetic effects propagate at the speed of light. In his paper [19] on propagation of waves in a compressible gas, the concept of a shock wave appears for the first time. Of incomparably greater importance for Riemann’s legacy is the doctoral thesis [14]. (It seems that Gauss, the formal thesis adviser, gave Riemann absolutely no help. Gauss’s remark in the thesis interview, that he had long been contemplating

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a publication on a related topic, must surely have given Riemann a fright!) The most important matters in the thesis going beyond Cauchy’s work are the idea of a Riemann surface, the treatment of complex functions as mappings, with a conformal property, the use of the “Dirichlet principle” (a minimization technique for existence theorems), and the famous “Riemann mapping theorem.” Not everything was rigorous, and much later work by others stems from trying to inject the desired rigor into the results. Riemann used the above ideas in several important papers, as we shall see below, quite apart from his application of complex analysis to the zeta function. Another hurdle remained before Riemann could teach in the university as a Privatdozent. This very junior position was, disagreeably, not salaried and the only income came from the fees paid by students who attended the Privatdozent’s lectures. There was no option, however, but to begin this way if Riemann wanted to be a university teacher. The obstacle that remained was the habilitation thesis. The topic, the theory of trigonometric series, was suggested by Dirichlet, who spent some time in Göttingen in the autumn of 1852. Dirichlet provided Riemann with references to previous work. Such references were dreadfully lacking in Riemann’s doctoral thesis; this was not apparently a grievous omission, as it would be today. Dirichlet and Riemann held some of their discussions at Weber’s house. In December 1853 Riemann turned in his thesis “On the representation of a function by a trigonometric series” to the dean, but because Gauss was in very poor health, it was not until Whitsun 1854 that the examination and trial lecture took place. The trial lecture was to become a landmark in the history of geometry, but we must first of all look at the habilitation thesis [21], which was also an important contribution to pure mathematics. The thesis was published posthumously by Dedekind from a complete manuscript in the “Nachlass,” the large body of miscellaneous writings left a Riemann’s death, and still available for study at the university library in Göttingen. Riemann was concerned in the thesis with necessary conditions for the representation of a function by a trigonometric series. His idea of using the function F obtained by two integrations of the trigonometric series was new and remains significant: see Zygmund [27, Chap. 9]. Riemann found it necessary to give a rigorous account of integration on the real line which was far superior to earlier work, in the thesis. Indeed, the Riemann integral is standard fare for undergraduates today. Other highlights included the “Riemann-Lebesgue lemma,” i.e., the vanishing at infinity of the Fourier coefficients of an integrable function; and the earliest use (it seems) of the principle of stationary phase in §13. The thesis was especially significant for its influence on Georg Cantor’s work, but that takes us well beyond Riemann’s lifetime. The “trial lecture” for Riemann’s habilitation, a colloquium of the philosophical faculty, was held on June 10, 1854. The title was “On the hypotheses on which geometry is based.” Riemann described a general mathematical object, an ndimensional differentiable manifold. The metric was based on a smoothly varying inner product on the tangent space (a “Riemannian metric”). He indicated that this was conceivably an accurate description of physical space (rather than the simplest case, Euclidean space), an idea the audience would have automatically detested. He

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went to considerable lengths to avoid formulas (there is only one in the published form of the whole paper). After being published from the Nachlass in 1868 [22], the lecture stimulated differential geometers for the rest of the century. Indeed, deep new results in Riemannian geometry continue to appear, such as Gromov’s “almost flat manifold theorem” of 1978 [8]. There is also a certain relation to the mathematical basis of the theory of general relativity, published by Einstein in 1915, which surely would have delighted Riemann. Readers might expect that after obtaining powerful new results in each of his two theses and the trial lecture, Riemann would have been invited to speak at conferences and seminars. This is to misunderstand the nature of nineteenth century mathematics research. Göttingen, one of the best universities in the world in the 1850s, had only one annual scientific meeting, which covered the natural sciences and medicine. Colloquium lectures by visiting scientists were virtually unknown. In mathematics, Felix Klein and Georg Cantor put forward the idea of an international congress only in the 1890s, the first one taking place in Zurich in 1897. Of course, there was some travel by individual mathematicians to meet colleagues in other states and countries; we shall see some more of this as the story continues. Richard Dedekind (1831–1916), Gauss’s last student, completed his habilitation just a few weeks after Riemann. At this stage, his work was at a rather pedestrian level, but it was not too long before his extraordinary talents came to the fore. He gave the first rigorous construction of the real numbers in 1858 using “Dedekind cuts,” although this was not published until 1872. His work on algebraic number fields is well known and of major importance, indeed its influence extends to abstract algebra as a whole. He was the first to prove the “Schröder-Bernstein theorem” of 1897, on comparability of infinite cardinal numbers. Dedekind obtained the result in 1887. It is remarkable that he was reluctant to publish such outstanding results. Going back to the 1850s, the two young instructors became friends, and the sociable and cheerful Dedekind was invaluable in lifting Riemann’s often plummeting spirits. We have records of the early courses taught by both scholars, and it seems that Galois theory was taught in a university for the first time by Dedekind in winter 1856/1857. Riemann, meanwhile, lectured regularly on his new complex function theory, among other topics. Riemann’s stock rose considerably when he published “Contributions to the theory of functions represented by the Gauss series F.˛; ˇ; ; x/” in the proceedings of the Göttingen Academy in 1857 [15]. He provided an entirely new approach to the hypergeometric function, studied by Gauss [7] and Kummer [10]. He specified the P-functions geometrically and used (in effect) groups of matrices to specify the behavior of P-functions under circuits of the branch points. All the results obtained via Kummer’s laborious derivations, much admired at the time, are obtained “almost without calculation” as Riemann remarks in an announcement of the paper [16]. The paper influenced later work of Fuchs and Hilbert, among others. Even more important for Riemann’s reputation was his treatment of a hot topic in “The theory of Abelian functions” in Borchardt’s Journal, 1857 [17]. The underlying idea is the study of the integrals of a rational function Z z R.x; y/dx 0

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where y is an algebraic function, say F.x; y/ D 0. Jacobi had shown in 1834 that if F is nonsingular of degree greater than 3, there will be more than two periods to such an integral. Weierstrass had treated F.x; y/ D y2  f .x/, deg f D 2n C 1, in papers of 1853 and 1856. Riemann used entirely different methods, requiring Riemann surfaces and the Dirichlet principle. See Weyl [26] for a rigorous account of Riemann’s results, an assessment of which could take us much too far afield here. In particular, there are key concepts in topology and algebraic geometry that derive their lineage from this crucial paper. The effort he put into it sent him into a state of mental exhaustion, and Dedekind describes in [4] how a walking trip in the Harz mountains that the two of them undertook restored Riemann’s spirits. It seems that they discussed Newton rather than pure mathematics. Dedekind deserves the gratitude of posterity for his efforts. On 23 February 1855, Gauss had died and Dirichlet had succeeded him in Göttingen. Riemann’s father and his sister Clara also died in this year. It was just as well that Riemann was given at this time a small stipend of 200 thalers a year from the government, since he had never been able to avoid accepting financial help from his father. Arguably the three best mathematicians in the world were together in Göttingen in the mid-1850s, but Dirichlet’s health gave way before too long and Dedekind was to leave for a professorship at the Polytechnic school in Zürich in 1858. (He spent the whole of his career from 1862 in his native Braunschweig.) In 1857, Riemann was promoted to extraordinary professor and his stipend increased to 300 thalers a year. But then his spirits were crushed by the death of his beloved brother Wilhelm, a clerk in the post office. Riemann now needed to provide for the upkeep of his three remaining sisters. (At this time the doors of the universities and professions were firmly closed to women, and the millions of women who worked only had access to menial jobs such as farm work, domestic service, and sweatshops [1].) After the sisters moved to Göttingen, the youngest, Marie, died in the spring of 1858. Nevertheless, Riemann gradually became happier and more self-confident living with family; he had always made trips back to see his family whenever he could. In the autumn of 1858, Riemann got to know the Italian mathematicians Francesco Brioschi (1824–1897), Enrico Betti (1823–1892), and Felice Casorati (1835–1890), who were on a tour that took in Paris, Berlin, and Göttingen. These men were hoping to build up Italian mathematics and intended to gain from their contacts with the best foreign mathematicians. Along with Eugenio Beltrami (1835– 1900), a student of Brioschi, they all played a significant part in the rise of Italian mathematics in the following years. This was also the starting point of Riemann’s contacts with Italy, which were so valuable to him in his last years. Dirichlet died in May 1859 after a long period of heart problems. Riemann was now deservedly appointed as his replacement and given living quarters at the Observatory. (The Göttingen Observatory retains its original appearance and since 2009 has housed the Lichtenberg-Kolleg Institute for advanced study in the humanities and social sciences.) Riemann was only thirty-two, and with good luck could have now looked forward to a prosperous and productive future beyond the end of the century. Sadly it was not to be. The reader may be astonished at the short

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lives of many of the characters in our narrative. It is well to note that for a man born in Germany in the 1870s, life expectancy was 36.5; for a woman, 38.5 [1]. Riemann’s lifespan of almost 40 years was not too bad when looked at in this way. In September 1859, Riemann visited the Berlin Academy of Sciences, which had recently elected him a corresponding member. Dedekind traveled with him, and he enjoyed a friendly welcome from the leading mathematicians Eduard Kummer (1810–1893), Carl Borchardt (1817–1880), Leopold Kronecker (1823–1891), and Karl Weierstrass (1815–1897). All of these will be familiar to the reader except perhaps Borchardt, who worked on the arithmetic–geometric mean problem (begun by Gauss) and continued Kummer’s work on the secular disturbances of the planets. In total he wrote 25 papers, as well as editing Crelle’s Journal from 1856 to 1880. Riemann’s visit was followed by the submission to the Berlin Academy of his paper [18] on zeta function and the frequency of prime numbers in October 1859. We defer discussion of this splendid paper until the end of our account of Riemann’s life. Like the Italian mathematicians, Riemann also visited Paris, spending a month there in 1860. This time he was able to meet the excellent scholars Joseph Serret (1819–1885), Joseph Bertrand (1822–1900), Charles Hermite (1822–1901), Victor Puiseux (1820–1883), Charles Briot (1817–1882), and Jean Claude Bouquet (1819– 1885). Of these, Joseph Bertrand may be the least familiar. He worked on analytical mechanics, thermodynamics, statistical probability, and the theory of curves and surfaces, as well as editing Journal des Savants (1865–1900). He also created “Bertrand’s postulate”: for any n > 3 there is a prime number p in .n; 2n  2/. This was proved by Chebyshev in 1852. Riemann was finally well enough off to contemplate marriage, and in June 1862 he married Elise Koch, a friend of his sisters. Now his health began to deteriorate, and she was his great support through his remaining 4 years. In July 1862 he had an attack of pleurisy, and this seems to have led to the lung disease that shortened his life. On the advice of Riemann’s doctor, Weber and the geology professor Sartorius von Waltershausen (1809–1876) pressed the authorities to give Riemann leave and financial support for a visit to Italy that would benefit his health. Riemann and Elise left in November 1862 and initially stayed with the consul in Messina, who was a friend of Sartorius. The long visit to Italy allowed him to visit many of the most famous cities and enjoy the art treasures and antiquities, as well as spending time in discussions with Enrico Betti in Pisa. Dedekind believed that Riemann enjoyed the greatest happiness of his life on his visits to Italy, partly because the people he encountered were so relaxed and informal. Not long after his return, his health was so bad that he began a second voyage to Italy in August 1863. When they were in Pisa, Elise gave birth to their daughter Ida. In May 1864 he took a villa in Pisa, and here at the end of August, his younger sister Helene died. He was offered an appointment at the University of Pisa, but felt his health was too precarious to accept. The next winter was passed in Pisa with his learned friends Betti, Beltrami, and others. He was now working on the memoir on the vanishing of theta functions [20]. In spite of very poor health, he went back to Göttingen, hoping to resume his teaching, arriving on 3 October 1865. His last

8

R. Baker

scientific work was a memoir on the mechanism of the ear, which he was unable to finish. On 15 June 1866, he and Elise set off to Italy. Their journey was interrupted at Cassel, where the war mentioned above had caused the destruction of the railway; but they did manage to complete the journey to Lake Maggiore, where he worked quietly, and enjoyed the scenery, until his death on 20 July, with Elise at his side. For further reading about Riemann’s life, see the book of Laugwitz [11] and the references therein.

2 The Zeta Function We now turn to Riemann’s memoir on the zeta function. Euler had given many results in number theory using product formulae; one of these was the formula for .s/ D

1 X

ns

.s 2 .1; 1//;

nD1

namely .s/ D

Y

.1  ps /1

.s > 1/:

p

He used this to show that

P

p1 diverges. Riemann viewed .s/ as a function of a

p

complex variable and used contour integration to show that 1

 2 s 



 Z 1  1 1 1 1 1 s .s/ D C x 2 s1 C x 2 s 2 !.x/dx 2 s.s  1/ 1

(1)

where !.x/ D

1 X

2 x

en

1

is related to a Jacobi theta function. Since the right side is unchanged if we replace s by 1  s, this gives both the analytic continuation of .s/ over the whole plane and the functional equation for .s/. We also easily obtain the set of zeros of .s/ in Re s < 0, namely f2; 4; 6; : : :g. Riemann gives a second proof of the functional equation. His next four assertions about .s/ are backed up by a number of indications of proof; Davenport [3] describes them as conjectures, but Riemann (as we know from his correspondence with Weierstrass) thought that just some details needed filling in.

An Introduction to Riemann’s Life, etc.

9

1. .s/ has infinitely many zeros in the critical strip. (These must clearly be placed symmetrically with respect to the real axis and with respect to the line Re s D 12 .) 2. The number N.T/ of zeros of .s/ in the “critical strip,” 0  Re s  1 with 0 < t  T satisfies N.T/ D

T T T log  C O.log T/: 2 2 2

It was not until 1905 that von Mangoldt gave the full proof of this (he had the result, with a worse error term, by 1895). 3. The entire function .s/ defined by   1 1 1 s 2 .s/ D s.s  1/  s .s/ 2 2 has the product representation ACBs

.s/ D e

 Y s s= e ; 1  

where A and B are constants. Here and below,  runs through the zeros of .s/ in the critical strip. In order to get a rigorous proof, Jacques Hadamard (1865– 1963), in the 1890s, developed the theory of entire functions of finite order, the distribution of their zeros, and their product representations. Hadamard’s proof of (3) dates to 1893. 4. We can write out an explicit formula for .x/  li x in terms of the zeros of .s/. This looks a lot tidier in the form proved by von Mangoldt in 1895. Let .n/ denote log p if n is a prime power pk , and .n/ D 0 otherwise. Then, for x 62 Z, X nx

.n/  x D 

X x  0 .0/ 1   log.1  x2 /;  .0/ 2 

where we must group  with N in the sum on the right. Riemann’s sketch proof involves an application of Fourier inversion but omits some key details. Riemann conjectures early in the paper that all complex zeros of .s/ lie on the “critical line” Re s D 12 . He says that after a few brief and fruitless attempts to prove this, he has put it on one side for the time being, because it did not seem to be essential to the immediate object of his investigation. Of course the Riemann hypothesis is now one of the best known unsolved problems in all of mathematics. We do know that more than 41% of the zeros of the zeta function up to a large height T are on the critical line [2]. Riemann makes the assertion [18] that he can prove this with asymptotically 100% in place of 41%; this still seems out of reach. It certainly seems on reading Riemann’s memoir that he regards the asymptotic formula (conjectured by Gauss) .x/  li x

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R. Baker

as a consequence of his investigation, but this theorem was only proved rigorously in 1895, by both Hadamard and Charles de la Vallée Poussin (1866–1962). A key element of the proof of this theorem, the “prime number theorem,” is the demonstration that no zero of .s/ has real part 1. Davenport [3] is the best source for the proofs of the results described above. There is a remarkable postscript to Riemann’s published work on the zeta function. In 1932 Carl Ludwig Siegel [24] published an account of material about the zeta function that he had found in the Nachlass. Bear in mind that Dedekind and Heinrich Weber had already combed through the Nachlass to find anything that could be included, as a coherent result, in their two editions of Riemann’s collected works! There are two topics covered in Siegel’s paper. One is an asymptotic formula for the computation of the real function Z.t/ D ei .t/ 



1 C it 2



  where .t/ is defined so that the sign of  12 C it is opposite to that of Z.t/. The other topic is a new representation of .s/ in terms of definite integrals. Riemann attained the formula Z.t/ D

N X nD1

 12

n

ei .t/Ci=4 et=2 2 cos. .t/  t log n/ C .2/1=2Cit .1  iet /

Z

1

CN

.x/ 2 Cit eNx dx ex  1

where CN is a certain path which begins and ends on the real axis at C1. This is amenable to the saddle point method; see Edwards [6, Chap. 9] for a discussion. As for the definite integral representation of .s/ mentioned above, this is said by Siegel to have been found in the Nachlass in 1926 by Erich Bessel-Hagen (1898–1946). Bessel-Hagen, a friend of Siegel, was then working on his habilitation at Göttingen. See [6, Chap. 7] for more details. We can certainly say that Riemann had obtained privately far more results about the zeta function than can be found in his memoir for the Berlin Academy, and that these led him to accurate empirical discoveries (such as the location of the first few zeros in the critical strip) that made his hypothesis on the zeta function plausible. For further study of the zeta function, besides the books of Davenport and Edwards mentioned above, there are interesting and well-written books by Titchmarsh [25], Ivic [9], and Patterson [13]. In summary, we can say that Riemann had tremendous insights in every branch of mathematics that he studied in any depth, and that he is quite an awe-inspiring figure. It is certainly still worthwhile to read his papers and to try and picture how his “fantastic premonitions” (as Dedekind describes them) looked to his contemporaries.

An Introduction to Riemann’s Life, etc.

11

References 1. D. Blackbourn, History of Germany 1780–1918, 2nd edn. (Blackwell, Oxford, 2003) 2. H.M. Bui, B. Conrey, M.P. Young, More than 41% of the zeros of the zeta function are on the critical line. Acta Arith. 150, 35–64 (2011) 3. H. Davenport, Multiplicative Number Theory, 3rd edn. (Springer, New York, 2000) 4. R. Dedekind, The Life of Bernhard Riemann. Collected Papers of Bernhard Riemann (translated by R.C. Baker, C. Christensen, H. Orde) (Kendrick Press, Heber City, 2004), pp. 518–533 5. P.G.L. Dirichlet, Lectures on Number Theory (with supplements by R. Dedekind, translated by J. Stillwell) (American Mathematical Society, Providence, RI/London Mathematical Society, London, 1999) 6. H.M. Edwards, Riemann’s Zeta Function (Academic, New York, 1974) 7. C.F. Gauss, Disquisitiones generales circa seriem infinitam, I and II. Werke, Vol. III (Georg Olms Verlag, Hildesheim, 1973/1812) 8. M. Gromov, Almost flat manifolds. J. Differ. Geom. 13, 223–230 (1978) 9. A. Ivic, The Riemann Zeta Function. Theory and Applications (Dover Publications, Inc., Mineola, NY, 2003) 10. E.E. Kummer, Über die Hypergeometrische Reihe. J. Reine Angew. Math. 15, 39–83 and 127–172 (1836); Collected Papers, Vol. II (Springer, Berlin/New York, 1975) 11. D. Laugwitz, Bernhard Riemann 1826–1866. Turning Points in the Conception of Mathematics (Birkhäuser Boston Inc., Boston, 2008) 12. A.-M. Legendre, Théorie des Nombres, 2nd edn. (1808). Reproduction, University of Michigan Library, Ann Arbor, MI 13. S.J. Patterson, An Introduction to the Theory of the Riemann Zeta Function (Cambridge University Press, Cambridge, 1988) 14. B. Riemann, Foundations for a General Theory of Functions of a Complex Variable. Collected Papers of Bernhard Riemann (translated by R.C. Baker, C. Christensen, H. Orde) (Kendrick Press, Heber City, 2004), pp. 1–41 15. B. Riemann, Contributions to the Theory of the Functions Represented by the Gauss Series F.˛; ˇ; ; x/. Collected Papers of Bernhard Riemann (translated by R.C. Baker, C. Christensen, H. Orde) (Kendrick Press, Heber City, 2004), pp. 57–75 16. B. Riemann, Author’s Announcement of the Preceding Paper. Collected Papers of Bernhard Riemann (translated by R.C. Baker, C. Christensen, H. Orde) (Kendrick Press, Heber City, 2004), pp. 77–78 17. B. Riemann, The Theory of Abelian Functions. Collected Papers of Bernhard Riemann (translated by R.C. Baker, C. Christensen, H. Orde) (Kendrick Press, Heber City, 2004), pp. 79–134 18. B. Riemann, On the Number of Primes Less than a Given Magnitude. Collected Papers of Bernhard Riemann (translated by R.C. Baker, C. Christensen, H. Orde) (Kendrick Press, Heber City, 2004), pp. 135–143 19. B. Riemann, On the Propagation of Planar Air Waves of Finite Amplitude. Collected Papers of Bernhard Riemann (translated by R.C. Baker, C. Christensen, H. Orde) (Kendrick Press, Heber City, 2004), pp. 145–165 20. B. Riemann, On the Vanishing of -Functions. Collected Papers of Bernhard Riemann (translated by R.C. Baker, C. Christensen, H. Orde) (Kendrick Press, Heber City, 2004), pp. 203–217 21. B. Riemann, On the Representation of a Function by a Trigonometric Series. Collected Papers of Bernhard Riemann (translated by R.C. Baker, C. Christensen, H. Orde) (Kendrick Press, Heber City, 2004), pp. 219–256 22. B. Riemann, The Hypotheses on Which Geometry is Based. Collected Papers of Bernhard Riemann (translated by R.C. Baker, C. Christensen, H. Orde) (Kendrick Press, Heber City, 2004), pp. 257–272

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23. B. Riemann, A Contribution to Electrodynamics. Collected Papers of Bernhard Riemann (translated by R.C. Baker, C. Christensen, H. Orde) (Kendrick Press, Heber City, 2004), pp. 273–278 24. C.L. Siegel, Über Riemanns Nachlass zur analytischen Zahlentheorie, Quellen und Studien zur Geschichte der Mathematik, Astronomie, und Physik, Abteilung B: Studien, Bd. 2, 45–80 25. E.C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd edn. (Oxford University Press, Oxford, 1986) 26. H. Weyl, The Concept of a Riemann Surface, 3rd edn. (translated by G.R. MacLane) (AddisonWesley, Boston, 1953) 27. A. Zygmund, Trigonometric Series, vols. I and II (Cambridge University Press, Cambridge, 1959)

Ramanujan’s Formula for .2n C 1/ Bruce C. Berndt and Armin Straub

Abstract Ramanujan made many beautiful and elegant discoveries in his short life of 32 years, and one of them that has attracted the attention of several mathematicians over the years is his intriguing formula for .2n C 1/. To be sure, Ramanujan’s formula does not possess the elegance of Euler’s formula for .2n/, nor does it provide direct arithmetical information. But, one of the goals of this survey is to convince readers that it is indeed a remarkable formula. In particular, we discuss the history of Ramanujan’s formula, its connection to modular forms, as well as the remarkable properties of the associated polynomials. We also indicate analogues, generalizations and opportunities for further research.

1 Introduction P s As customary, .s/ D 1 nD1 n ; Re s > 1, denotes the Riemann zeta function. Let Br , r  0, denote the r-th Bernoulli number. When n is a positive integer, Euler’s formula .2n/ D

.1/n1 B2n .2/2n 2.2n/Š

(1)

not only provides an elegant formula for evaluating .2n/, but it also tells us of the arithmetical nature of .2n/. In contrast, we know very little about the odd zeta values .2n C 1/. One of the major achievements in number theory in the past half-century is R. Apéry’s proof that .3/ is irrational [2], but for n  2, the arithmetical nature of .2n C 1/ remains open.

B.C. Berndt () Department of Mathematics, University of Illinois at Urbana–Champaign, 1409 W. Green St., Urbana, IL 61801, USA e-mail: [email protected] A. Straub Department of Mathematics and Statistics, University of South Alabama, 411 University Blvd N, Mobile, AL 36688, USA e-mail: [email protected] © Springer International Publishing AG 2017 H. Montgomery et al. (eds.), Exploring the Riemann Zeta Function, DOI 10.1007/978-3-319-59969-4_2

13

14

B.C. Berndt and A. Straub

Ramanujan made many beautiful and elegant discoveries in his short life of 32 years, and one of them that has attracted the attention of several mathematicians over the years is his intriguing formula for .2n C 1/. To be sure, Ramanujan’s formula does not possess the elegance of (1), nor does it provide any arithmetical information. But, one of the goals of this survey is to convince readers that it is indeed a remarkable formula. Theorem 1.1 (Ramanujan’s Formula for .2n C 1/) Let Br , r  0, denote the r-th Bernoulli number. If ˛ and ˇ are positive numbers such that ˛ˇ D  2 , and if n is a positive integer, then ˛

n

1 X 1 1 .2n C 1/ C 2nC1 .e2m˛  1/ 2 m mD1

.ˇ/

n

D 22n

!

1 X 1 1 .2n C 1/ C 2nC1 2 m .e2mˇ  1/ mD1

!

nC1 X B2nC22k B2k ˛ nC1k ˇ k : .1/k1 .2k/Š .2n C 2  2k/Š kD0

(2)

Theorem 1.1 appears as Entry 21(i) in Chap. 14 of Ramanujan’s second notebook [58, p. 173]. It also appears in a formerly unpublished manuscript of Ramanujan that was published in its original handwritten form with his lost notebook [59, formula (28), pp. 319–320]. The purposes of this paper are to convince readers why (2) is a fascinating formula, to discuss the history of (2) and formulas surrounding it, and to discuss the remarkable properties of the polynomials on the right-hand side of (2). We briefly point the readers to analogues and generalizations at the end of our paper. In Sect. 2, we discuss Ramanujan’s aforementioned unpublished manuscript and his faulty argument in attempting to prove (2). Companion formulas (both correct and incorrect) with the same parentage are also examined. In the following Sect. 3, we offer an alternative formulation of (2) in terms of hyperbolic cotangent sums. In Sects. 4 and 5, we then discuss a more modern interpretation of Ramanujan’s identity. We introduce Eisenstein series and their Eichler integrals, and observe that their transformation properties are encoded in (2). In particular, this leads us to a vast extension of Ramanujan’s identity from Eisenstein series to general modular forms. In a different direction, (2) is a special case of a general transformation formula for analytic Eisenstein series, or, in another context, a general transformation formula that greatly generalizes the transformation formula for the logarithm of the Dedekind eta function. We show that Euler’s famous formula for .2n/ arises from the same general transformation formula, and so Ramanujan’s formula (2) is a natural analogue of Euler’s formula. All of this is discussed in Sect. 6.

Ramanujan’s Formula for .2n C 1/

15

In Sect. 7, we discuss some of the remarkable properties of the polynomials that appear in (2). These polynomials have received considerable recent attention, with exciting extensions by various authors to other modular forms. We sketch recent developments and indicate opportunities for further research. We then provide in Sect. 8 a brief compendium of proofs of (2), or its equivalent formulation with hyperbolic cotangent sums.

2 Ramanujan’s Unpublished Manuscript The aforementioned handwritten manuscript containing Ramanujan’s formula for .2n C 1/ was published for the first time with Ramanujan’s lost notebook [59, pp. 318–321]. This partial manuscript was initially examined in detail by the first author in [12], and by Andrews and the first author in their fourth book on Ramanujan’s lost notebook [1, Chap. 12, pp. 265–284]. The manuscript’s content strongly suggests that it was intended to be a continuation of Ramanujan’s paper [55], [57, pp. 133–135]. It begins with paragraph numbered 18, giving further evidence that it was intended to be a part of [55]. One might therefore ask why Ramanujan did not incorporate this partial manuscript in his paper [55]. As we shall see, one of the primary claims in his manuscript is false. Ramanujan’s incorrect proof arose from a partial fraction decomposition, but because he did not have a firm grasp of the Mittag–Leffler Theorem, he was unable to discern his mis-application of the theorem. Most certainly, Ramanujan was fully aware that he had indeed made a mistake, and consequently he wisely chose not to incorporate the results in this manuscript in his paper [55]. Ramanujan’spmistake arose p when he attempted to find the partial fraction expansion of cot. w˛/ coth. wˇ/. We now offer Ramanujan’s incorrect assertion from his formerly unpublished manuscript. Entry 2.1 (p. 318, Formula (21)) If ˛ and ˇ are positive numbers such that ˛ˇ D  2 , then 1  X p p  mˇ coth.mˇ/ 1 m˛ coth.m˛/ D cot. w˛/ coth. wˇ/: C C 2 2 2w mD1 wCm ˛ wm ˇ 2

(3)

We do not know if Ramanujan was aware p of the Mittag–Leffler Theorem. p Normally, if we could apply this theorem to cot. w˛/ coth. wˇ/, we would let w ! 1 and conclude that the difference between the right- and left-hand sides of (3) is an entire function that is identically equal to 0. Of course, in this instance, we cannot make such an argument. We now offer a corrected version of Entry 2.1.

16

B.C. Berndt and A. Straub

Entry 2.2 (Corrected Version of (3)) Under the hypotheses of Entry 2.1, p p 1 ˇ  1 cot. w˛/ coth. wˇ/ D C log 2 2w 2 ˛  1 X m˛ coth.m˛/ mˇ coth.mˇ/ : C C w C m2 ˛ w  m2 ˇ mD1

(4)

Shortly after stating Entry 2.1, Ramanujan offers the following corollary. Entry 2.3 (p. 318, Formula (23)) If ˛ and ˇ are positive numbers such that ˛ˇ D  2 , then ˛

1 X

1 X m m ˛Cˇ 1 C ˇ D  : 2m˛  1 2mˇ  1 e e 24 4 mD1 mD1

(5)

To prove (5), it is natural to equate coefficients of 1=w on both sides of (3). When we do so, we obtain (5), but without the term  14 on the right-hand side of (5) [1, pp. 274–275]. Ramanujan surely must have been perplexed by this, for he had previously proved (5) by other means. In particular, Ramanujan offered Entry 2.3 as Corollary (i) in Sect. 8 of Chap. 14 in his second notebook [58], [10, p. 255]. Similarly, if one equates constant terms on both sides of (4), we obtain the familiar transformation formula for the logarithm of the Dedekind eta function. Entry 2.4 (p. 320, Formula (29)) If ˛ and ˇ are positive numbers such that ˛ˇ D  2 , then 1 X mD1

1 m.e2m˛

 1/



1 X mD1

1 m.e2mˇ

 1/

D

1 ˛ ˛ˇ log  : 4 ˇ 12

(6)

Of course, if we had employed (3) instead of (4), we would not have obtained the expression 14 log ˇ˛ on the right-hand side of (6). Entry 2.4 is stated by Ramanujan as Corollary (ii) in Sect. 8 of Chap. 14 in his second notebook [58], [10, p. 256] and as Entry 27(iii) in Chap. 16 of his second notebook [58], [11, p. 43]. In contrast to Entries 2.3 and 2.4, Ramanujan’s formula (2) can be derived from (3), because we equate coefficients of wn , n  1, on both sides, and so the missing terms in (3) do not come into consideration. We now give the argument likely given by Ramanujan. (Our exposition is taken from [1, p. 278].) Proof Return to (3), use the equality coth x D 1 C

e2x

2 ; 1

and expand the summands into geometric series to arrive at

(7)

Ramanujan’s Formula for .2n C 1/

17

(

  1 2 1 X  w k  2 1 C 2m˛ m kD0 m˛ e 1   ) 1  1X w k 2 :  1 C 2mˇ m kD0 m2 ˇ e 1 1

X p p  1 cot. w˛/ coth. wˇ/ D C 2 2w mD1

(8)

Following Ramanujan, we equate coefficients of wn , n  1, on both sides of (8). On the right side, the coefficient of wn equals .˛/n .2n C 1/ C 2.˛/n

1 X mD1

ˇ n .2n C 1/ C 2ˇ n

1 m2nC1 .e2m˛

 1/

1 X

1 : 2nC1 .e2mˇ  1/ m mD1

(9)

Using the Laurent expansions for cot z and coth z about z D 0, we find that on the left side of (8), 1 1 X p p 22j B2j  X .1/k 22k B2k  cot. w˛/ coth. wˇ/ D .w˛/k1=2 .wˇ/j1=2 : 2 2 kD0 .2k/Š .2j/Š jD0

(10) The coefficient of wn in (10) is easily seen to be equal to 2nC1

2

nC1 X kD0

.1/k

B2nC22k B2k ˛ k ˇ nC1k ; .2k/Š .2n C 2  2k/Š

(11)

where we used the equality ˛ˇ D  2 . Now equate the expressions in (9) and (11), then multiply both sides by .1/n 12 , and lastly replace k by n C 1  k in the finite sum. The identity (2) immediately follows. t u As mentioned in Sect. 1, Ramanujan also recorded (2) as Entry 21(i) of Chap. 14 in his second notebook [58, p. 173], [10, p. 271]. Prior to that on page 171, he offered the partial fraction decomposition 1 1 X X n coth.nx=y/ n coth.ny=x/  2xy ; 2 2 n Cy n2  x2 nD1 nD1 (12) which should be compared with (3). The two infinite series on the right side of (12) diverge individually, but when combined together into one series, the series converges. Thus, most likely, Ramanujan’s derivation of (2) when he recorded Entry 21(i) in his second notebook was similar to the argument that he used in his partial manuscript.

 2 xy cot.x/ coth.y/ D 1 C 2xy

18

B.C. Berndt and A. Straub

Sitaramachandrarao [69] modified Ramanujan’s approach via partial fractions to avoid the pitfalls experienced by Ramanujan. Sitaramachandrarao established the partial fraction decomposition 2 2 .y  x2 / (13) 3  1  2 X x2 coth.my=x/ y coth.mx=y/  2xy : C m.m2 C y2 / m.m2  x2 / mD1

 2 xy cot.x/ coth.y/ D 1 C

Using the elementary identities y2 1 m C D 2 2 2 2 m.m C y / m Cy m and m x2 1 D 2  ;  x2 / m  x2 m

m.m2

and then employing (7), he showed that 2 2 .y  x2 / (14) 3  1  X m coth.mx=y/ m coth.my=x/ C 2xy  m2 C y2 m2  x2 mD1

 2 xy cot.x/ coth.y/ D 1 C

 4xy

  1 X 1 1 1  : m e2mx=y  1 e2my=x  1 mD1

p p Setting x D w˛ and y D wˇ above and invoking the transformation formula for the logarithm of the Dedekind eta function from Entry 2.4, we can readily deduce (4). For more details, see [1, pp. 272–273]. The first published proof of (2) is due to Malurkar [49] in 1925–1926. Almost certainly, he was unaware that (2) can be found in Ramanujan’s notebooks [58]. If we set ˛ D ˇ D  and replace n by 2n C 1 in (2), we deduce that, for n  0, .4n C 3/ D 24nC2  4nC3

2nC2 X kD0

.1/kC1

1 X B4nC42k B2k k4n3 2 : .2k/Š .4n C 4  2k/Š e2k  1 kD1

(15)

Ramanujan’s Formula for .2n C 1/

19

This special case is actually due to Lerch [44] in 1901. The identity (15) is a remarkable identity, for it shows that .4n C 3/ is equal to a rational multiple of  4nC3 plus a very rapidly convergent series. Therefore, .4n C 3/ is “almost” a rational multiple of  4nC3 .

3 An Alternative Formulation of (2) in Terms of Hyperbolic Cotangent Sums If we use the elementary identity (7), we can transform Ramanujan’s identify (2) for .2n C 1/ into an identity for hyperbolic cotangent sums, namely,

˛ n

1 1 X X coth.˛m/ coth.ˇm/ n D .ˇ/ 2nC1 m m2nC1 mD1 mD1

 22nC1

nC1 X B2nC22k B2k ˛ nC1k ˇ k ; .1/k .2k/Š .2n C 2  2k/Š kD0

(16)

where, as before, n is a positive integer and ˛ˇ D  2 . To the best of our knowledge, the first recorded proof of Ramanujan’s formula (2) in the form (16) was by Nanjundiah [52] in 1951. If we replace n by 2n C 1 and set ˛ D ˇ D , then (16) reduces to 1 2nC2 X X B4nC42k coth.m/ B2k 4nC2 4nC3 : D 2  .1/kC1 4nC3 m .2k/Š .4n C 4  2k/Š mD1 kD0

(17)

This variation of (15) was also first established by Lerch [44]. Later proofs were given by Watson [72], Sandham [64], Smart [70], Sayer [65], Sitaramachandrarao [69], and the first author [7, 8]. The special cases n D 0 and n D 1 are Entries 25(i), (ii) in Chap. 14 of Ramanujan’s second notebook [58, p. 176], [10, p. 293]. He communicated this second special case in his first letter to Hardy [57, p. xxvi]. Significantly generalizing an idea of Siegel in deriving the transformation formula for the Dedekind eta function [68], Kongsiriwong [41] not only established (16), but he derived several beautiful generalizations and analogues of (16). Deriving a very general transformation formula for Barnes’ multiple zeta function, Komori, Matsumoto, and Tsumura [40] established not only (16), but also a plethora of further identities and summations in closed form for series involving hyperbolic trigonometric functions.

20

B.C. Berndt and A. Straub

4 Eisenstein Series A more modern interpretation of Ramanujan’s remarkable formula, which is discussed, for instance, in [30], is that (2) encodes the fundamental transformation properties of Eisenstein series of level 1 and their Eichler integrals. The goal of this section and the next is to explain this connection and to introduce these modular objects starting with Ramanujan’s formula. As in [26] and [30], set Fa .z/ D

1 X

a .n/e2inz ;

k .n/ D

X

nD1

dk :

(18)

djn

Observe that, with q D e2iz , we can express Fa .z/ as a Lambert series 0 1 1 1 X 1 1 1 X X X X X na qn na @ Fa .z/ D da A qn D da qdm D D 1  qn e2inz  1 nD1 dD1 mD1 nD1 nD1 djn

in the form appearing in Ramanujan’s formula (2). Indeed, if we let z D ˛i=, then Ramanujan’s formula (2) translates into 

   .2m C 1/ .2m C 1/ 1 C F2mC1 .z/ D z2m C F2mC1  2 2 z C

mC1 B2m2nC2 .2i/2mC1 X B2n z2n : 2z .2n/Š .2m  2n C 2/Š nD0

(19)

This generalization to values z in the upper half-plane H D fz 2 C W Im.z/ > 0g was derived by Grosswald in [25]. Ramanujan’s formula becomes particularly simple if the integer m satisfies m < 1. In that case, with k D 2m, Eq. (19) can be written as Ek .z/ D zk Ek .1=z/;

(20)

with 1

Ek .z/ D 1 C

X 2 2

k1 .n/qn : F1k .z/ D 1 C .1  k/ .1  k/ nD1

(21)

The series Ek , for even k > 2, are known as the normalized Eisenstein series of weight k. They are fundamental instances of modular forms. A modular form

Ramanujan’s Formula for .2n C 1/

21

of weight k and level 1 is a function G on the upper half-plane H , which is holomorphic on H (and as z ! i1), and transforms according to .cz C d/k G.Vz/ D G.z/

(22)

for each matrix

 ab VD 2 SL2 .Z/: cd Here, as usual, SL2 .Z/ is the modular group of integer matrices V with determinant 1, and these act on H by fractional linear transformations as Vz D

az C b : cz C d

(23)

Modular forms of higher level are similarly defined and transform only with respect to certain subgroups of SL2 .Z/. Equation (20), together with the trivial identity Ek .z C 1/ D Ek .z/, establishes that Ek .z/ satisfies the modular transformation (22) for the matrices



 0 1 11 SD ; TD : 1 0 01 It is well known that S and T generate the modular group SL2 .Z/, from which it follows that Ek .z/ is invariant under the action of any element in SL2 .Z/. In other words, Ek .z/ satisfies (22) for all matrices in SL2 .Z/. To summarize our discussion so far, the cases m < 1 of Ramanujan’s formula (2) express the fact that, for even k > 2, the q-series Ek .z/, defined in (21) as the generating function of sums of powers of divisors, transforms like a modular form of weight k. Similarly, the case m D 1 encodes the fact that E2 .z/ D 1  24

1 X

1 .n/qn ;

nD1

as defined in (21), transforms according to E2 .z/ D z2 E2 .1=z/ 

6 : iz

This is an immediate consequence of Ramanujan’s formula in the form (19). The weight 2 Eisenstein series E2 .z/ is the fundamental instance of a quasimodular form. In the next section, we discuss the case m > 0 of Ramanujan’s formula (2) and describe its connection with Eichler integrals.

22

B.C. Berndt and A. Straub

5 Eichler Integrals and Period Polynomials If f is a modular form of weight k and level 1, then   1  f .z/ D 0: zk f  z The derivatives of f , however, do not in general transform in a modular fashion by themselves. For instance, taking the derivative of this modular relation and rearranging, we find that   1 k  f 0 .z/ D f .z/: zk2 f 0  z z This is a modular transformation law only if k D 0, in which case the derivative f 0 (or, the 1st integral of f ) transforms like a modular form of weight 2. Remarkably, it turns out that any .k  1/st integral of f does satisfy a nearly modular transformation rule of weight k2.k1/ D 2k. Indeed, if F is a .k1/st primitive of f , then z

k2

  1  F.z/ F  z

(24)

is a polynomial of degree at most k  2. The function F is called an Eichler integral of f and the polynomials are referred to as period polynomials. Let, as usual, DD

1 d d Dq : 2i dz dq

Remark 5.1 That (24) is a polynomial can be seen as a consequence of Bol’s

identity [16], which states that, for all sufficiently differentiable F and V D ac db 2 SL2 .R/, .Dk1 F/.Vz/ D Dk1 Œ.cz C d/k2 F.Vz/: .cz C d/k

(25)

Here, Vz is as in (23). If F is an Eichler integral of a modular form f D Dk1 F of weight k, then Dk1 Œ.cz C d/k2 F.Vz/  F.z/ D

.Dk1 F/.Vz/  .Dk1 F/.z/ D 0; .cz C d/k

for all V 2 SL2 .Z/. This shows that .cz C d/k2 F.Vz/  F.z/ is a polynomial of degree at most k  2. A delightful explanation of Bol’s identity (25) in terms of Maass raising operators is given in [45, IV. 2].

Ramanujan’s Formula for .2n C 1/

23

Consider, for m > 0, the series F2mC1 .z/ D

1 X

2m1 .n/qn ;

nD1

which was introduced in (18) and which is featured in Ramanujan’s formula (19). Observe that 0 1 1 1 X X X @ D2mC1 F2mC1 .z/ D d2m1 A n2mC1 qn D

2mC1 .n/qn ; nD1

nD1

djn

and recall from (21) that this is, up to the missing constant term, an Eisenstein series of weight 2m C 2. It thus becomes clear that Ramanujan’s formula (19), in the case m > 0, encodes the fundamental transformation law of the Eichler integral corresponding to the weight 2m C 2 Eisenstein series. Remark 5.2 That the right-hand side of (19) is only almost a polynomial is due to the fact that its left-hand side needs to be adjusted for the constant term of the underlying Eisenstein series. To be precise, integrating the weight k D 2m C 2 Eisenstein series (21), slightly scaled, k  1 times, we find that G2mC1 .z/ WD

.1  2m/ .2iz/2mC1 C F2mC1 .z/ 2 .2m C 1/Š

(26)

is an associated Eichler integral of weight 2m. Keeping in mind the evaluation .1  2m/ D B2mC2 =.2m C 2/, Ramanujan’s formula in the form (19) therefore can be restated as   1 .2m C 1/ .1  z2m /  G2mC1 .z/ D z2m G2mC1  z 2 

m B2m2nC2 .2i/2mC1 X B2n z2n1 ; (27) 2 .2n/Š .2m  2n C 2/Š nD1

where the right-hand side is a polynomial of degree k  2 D 2m. Compare, for instance, [74, Eq. (11)]. An interesting and crucial property of period polynomials is that their coefficients encode the critical L-values of the corresponding modular form. Indeed, consider a modular form f .z/ D

1 X

a.n/qn

nD0

of weight k. For simplicity, we assume that f is modular on the full modular group SL2 .Z/ (for the case of higher level, we refer to [54]). Let

24

B.C. Berndt and A. Straub 1 X 1 a.n/ n a.0/zk1 C q .k  1/Š .2i/k1 nD1 nk1

f  .z/ WD

be an Eichler integral of f . Then, a special case of a result by Razar [62] and Weil [73] (see also [30, (8)]) shows the following. Theorem 5.3 If f and f  are as described above, then   k2 X L.k  1  j; f / j 1  f  .z/ D  zk2 f   z; z jŠ.2i/k1j jD0 where, as usual, L.s; f / D

P1

nD1

(28)

an ns .

Example 5.4 For instance, if " # 1 X 1 k1 k1 1 n .1  k/ C f .z/ D .2i/ .1  k/Ek .z/ D .2i/

k1 .n/q ; 2 2 nD1 where Ek is the Eisenstein series defined in (21), then f  .z/ D

1 zk1 .2i/k1 .1  k/ C Fk1 .z/ 2 .k  1/Š

equals the function G2mC1 .z/, where k D 2m C 2, used earlier in (26). Observe that the L-series of the Eisenstein series f .z/ is L.s; f / D .2i/k1

1 X

k1 .n/ nD1

ns

D .2i/k1 .s/.1  k C s/:

The result (28) of Razar and Weil therefore implies that   k2 X .2i/j .k  1  j/.j/ j 1  f  .z/ D  z; zk2 f   z jŠ jD0

(29)

and the right-hand side of (29) indeed equals the right-hand side of (27), which we obtained as a reformulation of Ramanujan’s identity. In particular, we see that (28) is a vast generalization of Ramanujan’s identity from Eisenstein series to other modular forms.

Ramanujan’s Formula for .2n C 1/

25

6 Ramanujan’s Formula for .2n C 1/ and Euler’s Formula for .2n/ are Consequences of the Same General Theorem Ramanujan’s formula (2) for .2n C 1/ is, in fact, a special instance of a general transformation formula for generalized analytic Eisenstein series or, in another formulation, for a vast generalization of the logarithm of the Dedekind eta function. We relate one such generalization due to the first author [5] and developed in [8], where a multitude of special cases

are derived. Throughout, let z be in the upper half-plane H , and let V D ac db be an element of SL2 .Z/. That is, a; b; c; d 2 Z and ad  bc D 1. Furthermore, let Vz D .az C b/=.cz C d/, as in (23). In the sequel, we assume that c > 0. Let r1 and r2 be real numbers, and let m be an integer. Define A.z; m; r1 ; r2 / WD

1 X X

km1 e2ik.nzCr1 zCr2 /

(30)

n>r1 kD1

and H.z; m; r1 ; r2 / WD A.z; m; r1 ; r2 / C .1/m A.z; m; r1 ; r2 /:

(31)

We define the Hurwitz zeta function .s; a/, for any real number a, by .s; a/ WD

X

.n C a/s ;

Re s > 1:

n>a

For real numbers a, the characteristic function is denoted by .a/. Define R1 D ar1 C cr2 and R2 D br1 C dr2 , where a; b; c; d are as above. Define   .s/  .r1 /.cz C d/s .eis .s; r2 / C .s; r2 // g.z; m; r1 ; r2 / WD lim s!m .2i/s C .R1 /..s; R2 / C eis .s; R2 // : (32) As customary, Bn .x/, n  0, denotes the n-th Bernoulli polynomial, and fxg denotes the fractional part of x. Let  WD fR2 gc  fR1 gd. Lastly, define h.z; m; r1 ; r2 / WD

c mC2 X X .1/k1 .cz C d/k1 jD1 kD0

kŠ.m C 2  k/Š

 Bk

   j  fR1 g jd C  BmC2k ; c c

where it is understood that if m C 2 < 0, then h.z; m; r1 ; r2 /  0. We are now ready to state our general transformation formula [5, p. 498], [8, p. 150].

26

B.C. Berndt and A. Straub

Theorem 6.1 If z 2 H and m is any integer, then .cz C d/m H.Vz; m; r1 ; r2 / D H.z; m; R1 ; R2 / C g.z; m; r1 ; r2 / C .2i/mC1 h.z; m; r1 ; r2 /: (33) We now specialize Theorem 6.1 by supposing that r1 D r2 D 0 and that Vz D 1=z, so that c D 1 and d D 0. Note that, from (30) and (31), H.z; m; 0; 0/ D .1 C .1/m /

1 X kD1

km1 : e2ikz  1

(34)

Theorem 6.2 If z 2 H and m is any integer, then zm .1 C .1/m /

1 X kD1

1 X km1 km1 m D .1 C .1/ C g.z; m/ / e2ik=z  1 e2ikz  1 kD1

C.2i/mC1

mC2 X kD0

Bk .1/ BmC2k .z/k1 ; kŠ .m C 2  k/Š (35)

where ( g.z; m/ D

i  log z;

if m D 0;

m

if m ¤ 0:

f1  .z/ g.m C 1/;

(36)

Corollary 6.3 (Euler’s Formula for .2n/) For each positive integer n, .2n/ D

.1/n1 B2n .2/2n : 2.2n/Š

(37)

Proof Put m D 2n  1 in (35). Trivially, by (34), H.z; 2m C 1; 0; 0/ D 0. Using (36), we see that (35) reduces to .1 C z2n1 /.2n/ D

.2/2n .1/n1 fB1 .1/B2n  B2n B1 z2n1 g; .2n/Š

(38)

where we have used the values Bk .1/ D Bk , k  2, and B2kC1 D 0, k  1. Since B1 .1/ D 12 and B1 D  12 , Euler’s formula (37) follows immediately from (38). u t Corollary 6.4 (Ramanujan’s Formula for .2n C 1/) Let ˛ and ˇ denote positive numbers such that ˛ˇ D  2 . Then, for each positive integer n,

Ramanujan’s Formula for .2n C 1/

( ˛

n

1

X k2n1 1 .2n C 1/ C 2 e2˛k  1

27

)

( D .ˇ/

n

kD1

22n

nC1 X

1

X k2n1 1 .2n C 1/ C 2 e2ˇk  1 kD1

.1/k

kD0

)

(39)

B2nC22k B2k ˛ nC1k ˇ k : .2k/Š .2n C 2  2k/Š

Proof Set m D 2n in (35), and let z D i=˛. Recall that  2 =˛ D ˇ. If we multiply both sides by 12 .ˇ/n , we obtain (39). t u We see from Corollaries 6.3 and 6.4 that Euler’s formula for .2n/ and Ramanujan’s formula for .2nC1/ are natural companions, because both are special instances of the same transformation formula. We also emphasize that in the foregoing proof, we assumed that n was a positive integer. Suppose that we had taken m D 2n, where n is a positive integer. Furthermore, if n > 1, then the sum of terms involving Bernoulli numbers is empty. Recall that .1  2n/ D 

B2n ; 2n

n  1:

(40)

We then obtain the following corollary. Corollary 6.5 Let ˛ and ˇ be positive numbers such that ˛ˇ D  2 . Then, for any integer n > 1, ˛n

1 1 X X k2n1 k2n1 B2n n  .ˇ/ D f˛ n  .ˇ/n g : 2˛k 2ˇk e  1 e  1 4n kD1 kD1

(41)

Corollary 6.5 is identical to Entry 13 in Chap. 14 of Ramanujan’s second notebook [58], [10, p. 261]. It can also be found in his paper [56, p. 269], [57, p. 190] stated without proof. Corollary 6.5 is also formula (25) in Ramanujan’s unpublished manuscript [59, p. 319], [1, p. 276]. Of course, by (40), we could regard (41) as a formula for .1  2n/, and so we would have a third companion to the formulas of Euler and Ramanujan. The first proof of (41) known to the authors is by Rao and Ayyar [60] in 1923, with Malurkar [49] giving another proof shortly thereafter in 1925. If we replace n by 2nC1 in (41), where n is a positive integer, and set ˛ D ˇ D , we obtain the special case 1 X B4nC2 k4nC1 D ; 2k  1 e 4.2n C 1/ kD1

(42)

which is due much earlier to Glaisher [24] in 1889. The formula (42) can also be found in Sect. 13 of Chap. 14 in Ramanujan’s second notebook [58], [10, p. 262].

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B.C. Berndt and A. Straub

There is yet a fourth companion. If we set m D 2 in (35), proceed as in the proof of Corollary 6.4, and use (40), we deduce the following corollary, which we have previously recorded as Entry 2.3, and which can be thought of as “a formula for .1/.” Corollary 6.6 Let ˛ and ˇ be positive numbers such that ˛ˇ D  2 . Then ˛

1 X kD1

1 X ˛Cˇ 1 k k C ˇ D  : 2ˇk  1 e2˛k  1 e 24 4 kD1

(43)

If ˛ D ˇ D , (43) reduces to 1 X kD1

k 1 1 D  : e2k  1 24 8

(44)

Both the special case (44) and the more general identity (43) can be found in Ramanujan’s notebooks [58, vol. 1, p. 257, no. 9; vol. 2, p. 170, Corollary 1], [10, pp. 255–256]. However, in 1877, Schlömilch [66], [67, p. 157] apparently gave the first proof of both (44) and (43). In conclusion of Sect. 6, we point out that several authors have proved general transformation formulas from which Ramanujan’s formula (39) for .2n C 1/ can be deduced as a special case. However, in most cases, Ramanujan’s formula was not explicitly recorded by the authors. General transformation formulas have been proved by Guinand [28, 29], Chandrasekharan and Narasimhan [18], Apostol [3, 4], Mikolás [50], Iseki [31], Bodendiek [14], Bodendiek and Halbritter [15], Glaeske [22, 23], Bradley [17], Smart [70], and Panzone, Piovan, and Ferrari [53]. Guinand [28, 29] did state (39). Due to a miscalculation, .2nC1/ did not appear in Apostol’s formula [3], but he later [4] realized his mistake and so discovered (39). Also, recall that in Sect. 3, we mentioned the very general transformation formula for multiple Barnes zeta functions by Komori, Matsumoto, and Tsumura [40] that contains Ramanujan’s formula (39) for .2n C 1/ as a special case. Lastly, in the beautiful work of Katsurada [38, 39] on asymptotic expansions of q-series, Ramanujan’s formula (2) arises as a special case. We also have not considered further formulas for .2n C 1/ that would arise from the differentiation of, e.g., (33) and (35) with respect to z; see [8].

7 The Associated Polynomials and Their Roots In this section we discuss the polynomials that are featured in Ramanujan’s formula (2) and discuss several natural generalizations. These polynomials have received considerable attention in the recent literature. In particular, several papers focus on the location of zeros of these polynomials. We discuss some of the

Ramanujan’s Formula for .2n C 1/

29

developments starting with Ramanujan’s original formula and indicate avenues for future work. Following [30] and [51], define the Ramanujan polynomials R2mC1 .z/ D

mC1 X nD0

B2m2nC2 B2n z2n .2n/Š .2m  2n C 2/Š

(45)

to be the polynomials appearing on the right-hand side of Ramanujan’s formula (2) and (19). The discussion in Sect. 5 demonstrates that the Ramanujan polynomials are, essentially, the odd parts of the period polynomials attached to Eisenstein series of level 1. In [51], Murty, Smyth, and Wang proved the following result on the zeros of the Ramanujan polynomials. Theorem 7.1 For m  0, all nonreal zeros of the Ramanujan polynomials R2mC1 .z/ lie on the unit circle. Moreover, it is shown in [51] that, for m  1, the polynomial R2mC1 .z/ has exactly four real roots and that these approach ˙2˙1 , as m ! 1. Example 7.2 As described in [51], one interesting consequence of this result is that there exists an algebraic number ˛ 2 H with j˛j D 1, ˛ 2m ¤ 1, such that R2mC1 .˛/ D 0 and, consequently,   F2mC1 .˛/  ˛ 2m F2mC1  ˛1 .2m C 1/ D ; 2 ˛ 2m  1

(46)

where F is as in (18). In other words, the odd zeta values can be expressed as the difference of two Eichler integrals evaluated at special algebraic values. An extension of this observation to Dirichlet L-series is discussed in [13]. Equation (46) follows directly from Ramanujan’s identity, with z D ˛, in the form (19). Remarkably, Eq. (46) gets close to making a statement on the transcendental nature of odd zeta values: Gun, Murty, and Rath prove in [30] that, as ˛ ranges over all algebraic values in H with ˛ 2m ¤ 1, the set of values obtained from the right-hand side of (46) contains at most one algebraic number. As indicated in Sect. 5, the Ramanujan polynomials are the odd parts of the period polynomials attached to Eisenstein series (or, rather, period functions; see Remark 5.2 and [74]). On the other hand, it was conjectured in [42], and proved in [43], that the full period polynomial is in fact unimodular, that is, all of its zeros lie on the unit circle. Theorem 7.3 For m > 0, all zeros of the polynomials R2mC1 .z/ C lie on the unit circle.

.2m C 1/ 2mC1 .z  z/ .2i/2mC1

30

B.C. Berndt and A. Straub

An analogue of Theorem 7.1 for cusp forms has been proved by Conrey, Farmer, and Imamoglu [19], who show that, for any Hecke cusp form of level 1, the odd part of its period polynomial has trivial zeros at 0, ˙2˙1 and all remaining zeros lie on the unit circle. Similarly, El-Guindy and Raji [20] extended Theorem 7.3 to cusp forms by showing that the full period function of any Hecke eigenform of level 1 has all its zeros on the unit circle. Remark 7.4 In light of (27) it is also natural to ask whether the polynomials pm .z/ WD

m B2m2nC2 .2m C 1/ .2i/2mC1 X B2n .1  z2m /  z2n1 2 2 .2n/Š .2m  2n C 2/Š nD1

are unimodular. Numerical evidence suggests that these period polynomials pm .z/ are indeed unimodular. Moreover, let p m .z/ be the odd part of pm .z/. Then, the polynomials p m .z/=z also appear to be unimodular. It seems reasonable to expect that these claims can be proved using the techniques used for the corresponding results in [51] and [43]. We leave this question for the interested reader to pursue. Very recently, Jin, Ma, Ono, and Soundararajan [32] established the following extension of the result of El-Guindy and Raji to cusp forms of any level. Theorem 7.5 For any newform f 2 Sk .0 .N// of even weight k and level N, all zeros of the p period polynomial, given by the right-hand side of (28), lie on the circle jzj D 1= N. Extensions of Ramanujan’s identity, and some of its ramifications, to higher level are considered in [13]. Numerical evidence suggests that certain polynomials arising as period polynomials of Eisenstein series again have all their roots on the unit circle. Especially in light of the recent advance made in [32], it would be interesting to study period polynomials of Eisenstein series of any level more systematically. Here, we only cite one conjecture from [13], which concerns certain special period polynomials, conveniently rescaled, and suggests that these are all unimodular. Conjecture 7.6 For nonprincipal real Dirichlet characters and ized Ramanujan polynomial

, the general-

  k X Bs; Bks; z  1 ks1 Rk .zI ; / D .1  zs1 / sŠ .k  s/Š M sD0

(47)

is unimodular, that is, all its roots lie on the unit circle. Here, Bn; are the generalized Bernoulli numbers, which are defined by 1 X

L

X .a/xeax xn ; Bn; D nŠ eLx  1 nD0 aD1

(48)

if is a Dirichlet character modulo L. If and are both nonprincipal, then Rk .zI ; / is indeed a polynomial. On the other hand, as shown in [13], if

Ramanujan’s Formula for .2n C 1/

31

k > 1, then R2k .zI 1; 1/ D R2k1 .z/=z, that is, despite their different appearance, the generalized Ramanujan polynomials reduce to the Ramanujan polynomials, introduced in (45), when D 1 and D 1. For more details about this conjecture, we refer to [13].

8 Further History of Proofs of (2) In this concluding section, we shall offer further sources where additional proofs of Ramanujan’s formula for .2n C 1/ can be found. We emphasize that two papers of Grosswald [25, 27], each proving formulas from Ramanujan’s second notebook, with the first giving a proof of (2), stimulated the first author and many others to seriously examine the content of Ramanujan’s notebooks [58]. Katayama [36, 37], Riesel [63], Rao [61], Vep˘stas [71], Zhang [75], and Zhang and Zhang [76] have also developed proofs of (2). Ghusayni [21] has used Ramanujan’s formula for .2n C 1/ for numerical calculations. For further lengthy discussions, see the first author’s book [10, p. 276], his survey paper [6], his account of Ramanujan’s unpublished manuscript [12], and his fourth book with Andrews [1, Chap. 12] on Ramanujan’s lost notebook. These sources also contain a plethora of references to proofs of the third and fourth companions to Ramanujan’s formula for .2n C 1/. Another survey has been given by Kanemitsu and Kuzumaki [33]. Also, there exist a huge number of generalizations and analogues of Ramanujan’s formula for .2n C 1/. Some of these are discussed in Berndt’s book [10, p. 276] and his paper [9]. Lim [46–48] has established an enormous number of beautiful identities in the spirit of the identities discussed in the present survey. Kanemitsu, Tanigawa, and Yoshimoto [34, 35], interpreting Ramanujan’s formula (2) as a modular transformation, derived further formulas for .2n C 1/, which they have shown lead to a rapid calculation of .3/ and .5/, for example. In Sects. 4 and 5 we have seen that Ramanujan’s formula can be viewed as describing the transformation laws of the modular Eisenstein series E2k , where k > 1, of level 1 (that is, with respect to the full modular group), the quasimodular Eisenstein series E2 as well as their Eichler integrals. We refer to [13] for extensions of these results to higher level.

References 1. G.E. Andrews, B.C. Berndt, Ramanujan’s Lost Notebook, Part IV (Springer, New York, 2013) 2. R. Apéry, Interpolation de fractions continues et irrationalite de certaines constantes. Bull. Section des Sci., Tome III (Bibliothéque Nationale, Paris, 1981), pp. 37–63 3. T.M. Apostol, Generalized Dedekind sums and transformation formulae of certain Lambert series. Duke Math. J. 17, 147–157 (1950) 4. T.M. Apostol, Letter to Emil Grosswald, January 24, 1973

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5. B.C. Berndt, Generalized Dedekind eta-functions and generalized Dedekind sums. Trans. Am. Math. Soc. 178, 495–508 (1973) 6. B.C. Berndt, Ramanujan’s formula for .2n C 1/, in Professor Srinivasa Ramanujan Commemoration Volume (Jupiter Press, Madras, 1974), pp. 2–9 7. B.C. Berndt, Dedekind sums and a paper of G.H. Hardy. J. Lond. Math. Soc. (2) 13, 129–137 (1976) 8. B.C. Berndt, Modular transformations and generalizations of several formulae of Ramanujan. Rocky Mt. J. Math. 7, 147–189 (1977) 9. B.C. Berndt, Analytic Eisenstein series, theta-functions, and series relations in the spirit of Ramanujan. J. Reine Angew. Math. 304, 332–365 (1978) 10. B.C. Berndt, Ramanujan’s Notebooks, Part II (Springer, New York, 1989) 11. B.C. Berndt, Ramanujan’s Notebooks, Part III (Springer, New York, 1991) 12. B.C. Berndt, An unpublished manuscript of Ramanujan on infinite series identities. J. Ramanujan Math. Soc. 19, 57–74 (2004) 13. B.C. Berndt, A. Straub, On a secant Dirichlet series and Eichler integrals of Eisenstein series. Math. Z. 284(3), 827–852 (2016). doi:10.1007/s00209-016-1675-0 14. R. Bodendiek, Über verschiedene Methoden zur Bestimmung der Transformationsformeln der achten Wurzeln der Integralmoduln k2 ./ und k02 ./, ihrer Logarithmen sowie gewisser Lambertscher Reihen bei beliebigen Modulsubstitutionen, Dissertation der Universität Köln, 1968 15. R. Bodendiek, U. Halbritter, Über die Transformationsformel von log ./ und gewisser Lambertscher Reihen. Abh. Math. Semin. Univ. Hambg. 38, 147–167 (1972) 16. G. Bol, Invarianten linearer Differentialgleichungen. Abh. Math. Semin. Univ. Hambg. 16, 1– 28 (1949) 17. D.M. Bradley, Series acceleration formulas for Dirichlet series with periodic coefficients. Ramanujan J. 6, 331–346 (2002) 18. K. Chandrasekharan, R. Narasimhan, Hecke’s functional equation and arithmetical identities. Ann. Math. 74, 1–23 (1961) 19. J.B. Conrey, D.W. Farmer, Ö. Imamoglu, The nontrivial zeros of period polynomials of modular forms lie on the unit circle. Int. Math. Res. Not. 2013(20), 4758–4771 (2013) 20. A. El-Guindy, W. Raji, Unimodularity of roots of period polynomials of Hecke eigenforms. Bull. Lond. Math. Soc. 46(3), 528–536 (2014) 21. B. Ghusayni, The value of the zeta function at an odd argument. Int. J. Math. Comp. Sci. 4, 21–30 (2009) 22. H.-J. Glaeske, Eine einheitliche Herleitung einer gewissen Klasse von Transformationsformeln der analytischen Zahlentheorie (I), Acta Arith. 20, 133–145 (1972) 23. H.-J. Glaeske, Eine einheitliche Herleitung einer gewissen Klasse von Transformationsformeln der analytischen Zahlentheorie (II). Acta Arith. 20, 253–265 (1972) 24. J.W.L. Glaisher, On the series which represent the twelve elliptic and the four zeta functions. Mess. Math. 18, 1–84 (1889) 25. E. Grosswald, Die Werte der Riemannschen Zeta-funktion an ungeraden Argumentstellen. Nachr. Akad. Wiss. Göttingen 9–13 (1970) 26. E. Grosswald, Remarks concerning the values of the Riemann zeta function at integral, odd arguments. J. Number Theor. 4(3), 225–235 (1972) 27. E. Grosswald, Comments on some formulae of Ramanujan. Acta Arith. 21, 25–34 (1972) 28. A.P. Guinand, Functional equations and self-reciprocal functions connected with Lambert series. Quart. J. Math. 15, 11–23 (1944) 29. A.P. Guinand, Some rapidly convergent series for the Riemann -function. Quart. J. Math. Ser. (2) 6, 156–160 (1955) 30. S. Gun, M.R. Murty, P. Rath, Transcendental values of certain Eichler integrals. Bull. Lond. Math. Soc. 43, 939–952 (2011) 31. S. Iseki, The transformation formula for the Dedekind modular function and related functional equations. Duke Math. J. 24, 653–662 (1957)

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58. S. Ramanujan, Notebooks, 2 vols. (Tata Institute of Fundamental Research, Bombay, 1957; 2nd ed., 2012) 59. S. Ramanujan, The Lost Notebook and Other Unpublished Papers (Narosa, New Delhi, 1988) 60. M.B. Rao, M.V. Ayyar, On some infinite series and products. Part I. J. Indian Math. Soc. 15, 150–162 (1923/1924) 61. S.N. Rao, A proof of a generalized Ramanujan identity. J. Mysore Univ. Sect. B 28, 152–153 (1981–1982) 62. M.J. Razar, Values of Dirichlet series at integers in the critical strip, in Modular Functions of One Variable VI. Lecture Notes in Mathematics, vol. 627, ed. by J.-P. Serre, D.B. Zagier (Springer, Berlin/Heidelberg, 1977), pp. 1–10 63. H. Riesel, Some series related to infinite series given by Ramanujan. BIT 13, 97–113 (1973) 64. H.F. Sandham, Some infinite series. Proc. Am. Math. Soc. 5, 430–436 (1954) 65. F.P. Sayer, The sum of certain series containing hyperbolic functions. Fibonacci Quart. 14, 215–223 (1976) 66. O. Schlömilch, Ueber einige unendliche Reihen. Ber. Verh. K. Sachs. Gesell. Wiss. Leipzig 29, 101–105 (1877) 67. O. Schlömilch, Compendium der höheren Analysis. zweiter Band, 4th ed. (Friedrich Vieweg und Sohn, Braunschweig, 1895) p 68. C.L. Siegel, A simple proof of .1=/ D ./ =i. Mathematika 1, 4 (1954) 69. R. Sitaramachandrarao, Ramanujan’s formula for .2n C 1/, Madurai Kamaraj University Technical Report 4, pp. 70–117 70. J.R. Smart, On the values of the Epstein zeta function. Glasgow Math. J. 14, 1–12 (1973) 71. L. Vep˘stas, On Plouffe’s Ramanujan identities. Ramanujan J. 27, 387–408 (2012) 72. G.N. Watson, Theorems stated by Ramanujan II. J. Lond. Math. Soc. 3, 216–225 (1928) 73. A. Weil, Remarks on Hecke’s lemma and its use, in Algebraic Number Theory: Papers Contributed for the Kyoto International Symposium, 1976, ed. by S. Iyanaga (Japan Society for the Promotion of Science, Tokyo, 1977), pp. 267–274 74. D.B. Zagier, Periods of modular forms and Jacobi theta functions. Invent. Math. 104, 449–465 (1991) 75. N. Zhang, Ramanujan’s formula and the values of the Riemann zeta-function at odd positive integers (Chinese). Adv. Math. Beijing 12, 61–71 (1983) 76. N. Zhang, S. Zhang, Riemann zeta function, analytic functions of one complex variable. Contemp. Math. 48, 235–241 (1985)

Towards a Fractal Cohomology: Spectra of Polya–Hilbert Operators, Regularized Determinants and Riemann Zeros Tim Cobler and Michel L. Lapidus

Abstract Emil Artin defined a zeta function for algebraic curves over finite fields and made a conjecture about them analogous to the famous Riemann hypothesis. This and other conjectures about these zeta functions would come to be called the Weil conjectures, which were proved by Weil in the case of curves and eventually, by Deligne in the case of varieties over finite fields. Much work was done in the search for a proof of these conjectures, including the development in algebraic geometry of a Weil cohomology theory for these varieties, which uses the Frobenius operator on a finite field. The zeta function is then expressed as a determinant, allowing the properties of the function to relate to the properties of the operator. The search for a suitable cohomology theory and associated operator to prove the Riemann hypothesis has continued to this day. In this paper we study the properties of the derivative operator D D dzd on a particular family of weighted Bergman spaces of entire functions on C. The operator D can be naturally viewed as the “infinitesimal shift of the complex plane” since it generates the group of translations of C. Furthermore, this operator is meant to be the replacement for the Frobenius operator in the general case and is used to construct an operator associated with any given meromorphic function. With this construction, we show that for a wide class of meromorphic functions, the function can be recovered by using a regularized determinant involving the operator constructed from the meromorphic function. This is illustrated in some important special cases: rational functions, zeta functions of algebraic curves (or, more generally, varieties) over finite fields, the Riemann zeta function, and culminating in a quantized version of the Hadamard factorization theorem that applies to any entire function of finite order. This shows that all of the information about the given meromorphic function is encoded into the special operator we constructed. Our construction is motivated in part by work of Herichi and the second author on the infinitesimal shift of the real line (instead of the complex plane) and the associated spectral operator, as well as by earlier work and

T. Cobler Department of Mathematics, Fullerton College, Fullerton, CA 92832, USA e-mail: [email protected] M.L. Lapidus () Department of Mathematics, University of California, Riverside, CA 92521, USA e-mail: [email protected] © Springer International Publishing AG 2017 H. Montgomery et al. (eds.), Exploring the Riemann Zeta Function, DOI 10.1007/978-3-319-59969-4_3

35

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conjectures of Deninger on the role of cohomology in analytic number theory, and a conjectural “fractal cohomology theory” envisioned in work of the second author and of Lapidus and van Frankenhuijsen on complex fractal dimensions.

1 Introduction Riemann’s famous paper [31] opened up the use of complex analysis to study the prime numbers. This approach has yielded many great results in number theory, including, but certainly not limited to, the Prime Number Theorem. Riemann also made his well-known conjecture that stands to this very day. We will refer to this as (RH) in this paper. Conjecture (RH) The only nontrivial zeros of .s/ occur when s satisfies 0. Then again using (13), we see that there is t0 > 0 such that .t/  .a C /t for t  t0 . Thus by part 2 of Theorem 8 we have that kDn k  CnŠrn e.aC/r for any r > 0, n D 1; 2; : : :, where C is a constant depending only on . Minimizing this expression with respect to r nyieldsn the critical n value r D aC . Substituting this choice of r gives kDn k  C nŠe .aC/ . Applying nn Stirling’s formula gives that kDn k  f .n/, where f .n/ is asymptotic to a constant p 1 times n.a C /n . Thus we have r.D/ D lim kDn k n  a C . We conclude that n!1

r.D/ D a and .D/ D a , as desired. u t To further study this operator, we restrict our attention to the special case when pR D 2, where we actually have a Hilbert space with inner product given by .f ; g/ D 2.jzj/ d . It is convenient to have a particular simple orthonormal basis to R2 f ge deal with, and, since we are dealing with entire functions that are guaranteed to have convergent power series, it makes sense to look at polynomials to try to find this orthonormal basis. It turns out that all we need are monomials. Theorem 10 ([2]) There exist constants cn such that fun .z/g, where un .z/ D cn zn , for n 2 N0 , forms an orthonormal basis for B2w . Proof First note that .zn ; zm / D 0 if n ¤ m. This follows from a simple calculation using the fact that the weight function is radial, so that using polar coordinates, we have .zn ; zm / D

Z

2

Z

0

Z D D0

rn ein rm eim e2.r/ rdrd

0

Z

2 i.mn/

e 0

1

d 0

1

rnCmC1 e2.r/ dr

Regularized Determinants and Riemann Zeros

47

if n ¤ m. Note that the integral on r converges for any n; m 2 N0 by the properties of our weight function. Thus the monomials form an orthogonal set. This orthogonal set is complete because every entire function has a convergent power series on C. Thus, if we choose cn D kz1n k , we normalize our set and, hence, the resulting sequence fun g is an orthonormal basis for B2w . u t Now we specialize further by choosing the family of weight functions given by ˛ w.z/ D ejzj for ˛ 2 R with 0 < ˛  1. (Note that in the notation of (13) and of Theorem 9 above, we then have a D 1 if ˛ D 1 and a D 0 if 0 < ˛ < 1.) We will call the resulting Hilbert space H˛ WD B2w . In this case, we can actually find the constants cn explicitly. Theorem 11 ([2]) If 0 < ˛  1, then for n 2 N0 , we have that kzn k2H˛ D

2  2 .nC1/ 2 ˛  ˛



 2 .n C 1/ : ˛

Proof Computing the norm in H˛ gives kzn k2H˛ D

Z

Z

˛

jzjn e2r dz D

Z

2

1d

C

0

1

˛

r2nC1 e2r dr:

0

For the integral over r, we make the change of variable x D 2r˛ , which changes the integral into kzn k2H˛

Z

1

D 2 0

x1

!2nC1

˛

ex

2

1 2˛. 2x /

˛1 ˛

dx D

2  2 .nC1/ 2 ˛ ˛

Z

1

x

2.nC1/ 1 ˛

ex dx:

0

However, the final integral is simply  . ˛2 .n C 1//. u t Thus we can simply take the normalizing constants cn to be the square root of the reciprocal of the formula for kzn k2 given above. Namely,  cn D

2 ˛

 12 2

nC1 ˛

 12   2 .n C 1/  ˛

for every n 2 N0 . Now examining the action of D on a typical basis element un .z/, n we see that: Dun D D.cn zn / D ncn zn1 D cnc un1 . We thus obtain the following n1 representation of D: Theorem 12 The operator D is isomorphic to a weighted backward shift on H˛ P1 taking a sequence of coefficients .an / in l2 .C/, where f .z/ D nD0 an un .z/, to .n anC1 / in l2 .C/, where for n 2 N0 , n > 0 and n is given by n2 D 2

2˛

.nC1/2  . ˛2 .nC1// .  . ˛2 .nC2//

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Proof Using the last calculation and writing f .z/ D Df .z/ D

1 X nD0

where n D

1

an

nD0

an un , we obtain 1

X .n C 1/cnC1 X ncn un1 D anC1 un D n anC1 un ; cn1 cn nD0 nD0

.nC1/cnC1 . It follows, using the cn

n2 D

P1

previously calculated formula for cn , that

2 2 .n C 1/2 c2nC1 .n C 1/2 kzn k2 2 .n C 1/  . .n C 1// ˛ ˛ D D 2 ; c2n kznC1 k2  . ˛2 .n C 2//

as desired. u t The last fact we will need from [2] is to apply the standard asymptotic for the 1 Gamma function to obtain that n  c  n1 ˛ as n ! 1, where c is a positive constant. Thus if 0 < ˛ < 1, then n ! 0 as n ! 1. Continuing beyond the results from [2], we start by calculating the adjoint D . P Theorem 13 Given f 2 H˛ , let f D 1 nD0 an un be its expansion in terms of the orthonormal basis. The adjoint of D is isomorphic to a weighted forward shift given by the equation D .an / D .n1 an1 /. P1 Proof To calculate D , write D f D nD0 bn un . Since fun g is an orthonormal basis we find the nth, for n  1, coefficient of D f : .D f ; un / D .f ; Dun / D .f ; n1 un1 / D n1 an1 . Thus we have bn D n1 an1 for each n  1. For b0 , we calculate .D f ; u0 / D .f ; Du0 / D .f ; 0/ D 0. Thus, D acts on the sequence of coefficients .an / as a weighted forward shift .an / 7! .n1 an1 /, with the new 0th coefficient being 0. u t Now that we have the adjoint, we can immediately see that D is not self-adjoint as D is a backward shift and D is a forward shift. Moreover, the following calculation with f .z/  1 shows that it is not even normal: Indeed, D Df D D 0 D 0, but, on the other hand, DD f D D.0 z/ D 0 . This shows that we cannot apply the functional calculus for unbounded normal operators that was used in [20–23]. Instead we use the Riesz functional calculus, which is valid for bounded operators like D. 1 Next, we will use the asymptotic n  c  n1 ˛ to determine which trace ideals D will belong to, depending on ˛. Theorem 14 The operator D is compact on H˛ for any 0 < ˛ < 1, trace class for any 0 < ˛ < 12 , Hilbert–Schmidt for any 0 < ˛ < 23 , and, in general, D 2 Jp if p ˛ < pC1 for any p 2 N, where Jp is the trace ideal defined in the previous section. Proof Let 0 N f n1 g.

Regularized Determinants and Riemann Zeros

49

1

But n  c  n1 ˛ ! 0, as n ! 1, for any 0 < ˛ < 1. Thus EN converges D is compact. Furthermore, we can write P to D in norm and therefore  DD 1 when nD1 n1 .un ; /un1 so n1 are the singular values of D. To P determine n1 are in lp , we use the Limit Comparison Test to compare 1 . n1 /p with nD1   P1 1 ˛1 p / , which converges if and only if p 1  ˛1 < 1. Solving this gives nD1 .n p p . Therefore D 2 Jp if ˛ < pC1 and, in particular, is trace class if p < 12 and ˛ < pC1 2 Hilbert–Schmidt if p < 3 . (Here, we have used the notation of Definition 5.) u t 1 From now on, we will fix an ˛ with 0 < ˛ < 2 and simply refer to H˛ as H. In this case, we have the following spectrum for D. Theorem 15 We have .D/ D p .D/ D f0g and 0 is a simple eigenvalue of D with eigenfunction f .z/  1, the constant function equal to 1. .t/ , as in (13). t t Here we have .t/ D t˛ for 0 < ˛ < 12 . Thus we have that lim D lim t˛1 D 0. t!1 t t!1 It follows that a D 0 and hence, by Theorem 9, .D/ D 0 D f0g. However, we also know that f .z/  1 2 H, so that D has the eigenvector f corresponding to the eigenvalue 0 and the point spectrum of D is also p .D/ D f0g. u t Finally, we turn to considering the set of operators fesD gs2C . We compare this to the result for @c obtained in [23] and mentioned in Sect. 1. This theorem will show that D is the infinitesimal shift (of the complex plane). Proof We know from Theorem 9 that .D/ D a , where a D lim

t!1 ˛

Theorem 16 The family fesD gs2C gives the group of translation operators on H. Proof First note that since any f 2 H is an entire function, we have the convergent P f .n/ .z/ n power series representation: f .z  s/ D 1 nD0 nŠ .s/ for any z; s 2 C. Thus esD f .z/ D

1 1 X X 1 1 dn .sD/n f .z/ D .s/n n f .z/ D f .z  s/: nŠ nŠ dz nD0 nD0

This shows that esD just acts as translation by s on the space H. From this expression we also see that lim kesD f  f k D 0. u t s!0

4 The Construction With the results about D in hand, we turn to constructing an operator that might play the role of Frobenius when dealing with the Riemann zeta function or other entire or meromorphic functions of interest in number theory, analysis, or mathematical physics.

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We begin by considering a particular choice of the family of analytic functions  .z/ D z C  . This gives us operators D WD  .D/ D D C I;

(18)

for which the following lemma holds. (Recall that ˛ has been fixed once for all to satisfy 0 < ˛ < 12 and hence, that Theorem 15 applies.) Lemma 2 For any  2 C, D 2 B.H/ with spectrum .D / D fg. If  ¤ 0, then D is invertible and D1  2 B.H/. Proof Applying the functional calculus on bounded operators along with the Spectral Mapping Theorem to the operator D and the function  .z/ gives a bounded operator D with spectrum .D / D  .f0g/ D fg, where we have used Theorem 15 according to which .D/ D f0g. Furthermore, if  ¤ 0, then 0 … .D / and it follows that D has a bounded inverse. u t This gives us a family of operators, each of whose spectra are each a single point, which can be any complex number. Recall that in the situation of the cohomology theory that helped prove the Weil conjectures, we would like an operator whose eigenvalues on different cohomology spaces are the zeros and poles of the zeta function we are interested in. In order to obtain an operator whose spectrum can represent the zero or pole set of a meromorphic function, we use the following construction. If Z D fz1 ; z2 ; : : :g is a (finite or countable) multiset of complex numbers, let Hn be a copy of the weighted Bergman space H and associate an operator Dn to be Dzn on Hn . (Here and thereafter, a multiset is a set with integer L multiplicities.) Finally, define the Hilbert space H D H with operator DZ D Z n n L n Dn . This gives Theorem 17 For Z D fz1 ; z2 ; : : :g, the operator DZ , constructed above, has spectrum .DZ / D fz1 ; z2 ; : : :g. Furthermore, for each i 2 N, zi is an eigenvalue of DZ and the number of linearly independent eigenvectors of zi for DZ in HZ is equal to the number of times zi occurs in the multiset Z D fz1 ; z2 ; : : :g. Proof For each n 2 N, let en 2 HZ be the element which is the constant function, with value 1 in the nth component, and 0 in every other component. Then DZ en D zn en and so zn is an eigenvalue with eigenvector en . Suppose zn1 D zn2 D    D znk D z. Then z is an eigenvalue with eigenvectors en1 ; en2 ; : : : enk and so there are at least as many linearly independent eigenvectors of z for DZ as the multiplicity of z in the multiset. Next, recall that the only eigenvalue of dzd on H is 0. Thus, the only eigenvalue of Dzn is zn . Suppose now that DZ x D zx for some x, we must either have the nth component of x being 0 or z D zn and so there cannot be any more linearly independent eigenvectors of z for DZ . Now that we know zi is an eigenvalue of DZ for each i, we know that fz1 ; z2 ; : : :g .DZ /. Next, let 2 C  fz1 ; z2 ; : : :g. Then

Regularized Determinants and Riemann Zeros

d D infn0 j  zn j > 0. Since D D

51

d dz

is quasinilpotent, r.D/ D 0 and so there is a  k positive integer N such that for every integer k  N, we have kDk k < d2 . Then, on P Dk the nth component of HZ , we have that 1 kD0 j zn j is absolutely convergent because 1 1  d k X X kDk k 1 2  D N1 : k j  zn j kDN d 2 kDN

Then we can calculate the inverse on the nth component via the absolutely convergent series: . I  Dzn /1 D ..  zn /I  D/1 D

1 1 X Dk :  zn kD0  zn

Further, by the same estimate k. I  Dzn /1 k  C uniformly in n, where CD

N X kDk k kD0

dk

C 21N :

(19)

L Therefore, n . I  Dzn /1 2 B.HZ / and so . I  DZ /1 exists and is bounded. That is, 2 .DZ /, the resolvent set of DZ ; recall that by definition, .DZ / is the complement of .DZ / in C. Hence, .DZ / D fz1 ; z2 ; : : :g. u t Corollary 1 If Z D fz1 ; z2 ; : : :g is either the zero or pole set of a meromorphic function f .z/, then Z is a discrete set and so we have exactly .DZ / D Z counting multiplicity. Moreover, each zi in Z is an eigenvalue of DZ , with multiplicity equal to the multiplicity of zi in the multiset Z. Thus, DZ has all of the information from the multiset fz1 ; z2 ; : : :g contained in its spectrum. If we then consider the multiset to be the zeros and poles of a meromorphic function f .z/, then the operator DZ contains these pieces of information of this function. This was, in fact, the original goal of this direction. In [23] (also [20, 21], and [22]) the spectrum of the operators studied there were vertical lines in the complex plane and the values of .s/ on these lines. This work was meant to approach the problem in a similar, but different, way and isolate out the zeros and poles of certain meromorphic functions. Theorem 17 gives a positive result that we have created such an operator. However, we wanted to go further and find a way to use determinant formulas to fully recover all of the values of the desired function as was done in the Weil conjectures. Unfortunately, we cannot use the determinant formulas for operators given in Sect. 2 for the operator DZ to recover f .z/ as a whole, because with this formulation DZ is not trace class. Even looking at just a single one of the terms, D D D C  I, with  ¤ 0, is not compact, let alone trace class (or, consequently, in any of the operator ideals Jp ), because our Hilbert space is infinite-dimensional. (Indeed, in a Banach space, the identity operator, like the closed unit ball, is compact if and only if the space is finite-dimensional.)

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The first such modification we will make is to restrict each D to its eigenspace, E, the space of constant functions. This restriction is then a compact operator. We also need to make a second adjustment from the original idea. Any zero or pole set of a meromorphic function will be a discrete set, and hence if there are infinitely many zeros or poles, they must tend to 1. This would then imply that the operator DZ described here is unbounded. What allows us to repair this problem and simultaneously recover the given function f .z/ using determinants, is to have our set Z consist of the reciprocals of the zeros rather than of the zeros themselves, and similarly for the poles. (In each case, the multiplicities of the zeros or the poles are taken into account.) The set of reciprocals will not necessarily be discrete as if the set is infinite, the sequence will tend to 0. However, rather than being a problem, this is actually completely required. Indeed, a compact operator on an infinite-dimensional space cannot have a bounded inverse and so 0 must be in the spectrum of any such compact operator. Combining this observation with the fact that regularized determinants apply only to trace ideals of compact operators, we see that having 0 in the closure of the set of reciprocals is necessary to apply the determinant theory to DZ . One comment to make about the restriction of the operator DZ to its total eigenspace, EZ , is that it simplifies the operator to a multiplication operator because the derivative on constant functions is just the zero operator. This is unfortunate as we do lose some of the rich setting of the Bergman space that has been used thus far. We are currently exploring alternative constructions in [5] that would allow us to remove this restriction and work on all of H. However, as we will see in the next section, the new version of the operator obtained by restriction will retain the desired spectrum from Theorem 17. In addition, we observe that in some sense, by analogy with what happens for curves over finite fields for Weil-type cohomologies and with what is expected in more general situations associated with the Riemann zeta function and other L-functions (see, e.g., [10, 12] and [25, esp. Appendix B]), EZ (the total eigenspace of DZ ) is the counterpart in our context of the total cohomology space (or, in the terminology of [25, 28], the total “fractal cohomology space”) and correspondingly, the restriction of the original (generalized) Polya–Hilbert operator DZ to its eigenspace EZ is the counterpart of the linear endomorphism induced by the Frobenius morphism on the (total) cohomology space. (See Sect. 1.1 above.) Therefore, this modification of the original operator seems natural (and perhaps unavoidable) in order to obtain a suitable determinant formula, of the type obtained in Sects. 4.1 and 5 below.

4.1 Refining the Operator of a Meromorphic Function First, let Z D .zn / be a sequence of complex numbers. Let Dn WD D C zn I be the operator in the previous section restricted functions E. Lto the subspace of constant L (It is clear that Dn is normal.) Let DZ D n Dn act on the space EZ D n E which

Regularized Determinants and Riemann Zeros

53

is a closed subspace of the Hilbert space HZ from the last section. So in actuality, this new definition of DZ is just the restriction to the Hilbert space EZ of the operator given in the previous section. First we note that this restriction still retains the main property from the last section. Theorem 18 For each n 2 N, each zn is an eigenvalue of DZ and the number of linearly independent eigenvectors associated with zn is equal to the number of times zn occurs in the sequence Z. Furthermore, .DZ / D fzn W n D 1; 2; 3; : : :g. Proof Let en be the eigenfunctions from the previous proof. Then since en is constant in each coordinate, en 2 EZ . Thus when restricted to the space of functions constant on each coordinate, DZ retains all of its eigenvalues and eigenvectors from before. Finally, we note that .Dzn / D fzn g from which it follows that

.DZ / D fzn W n D 1; 2; 3; : : :g as in the proof of Theorem 17. u t Remark 1 Note that in the case when Z D .zn / is the sequence of the reciprocals of the nonzero elements in the zero set or the pole set of a given meromorphic S function (as in Sect. 5 below), then .DZ / D Z if Z is finite and .DZ / D Z f0g if Z is infinite. The next theorem shows that this restriction of the operator will truly give us what we need for our quantized number theory framework. Theorem 19 We have the following relationships between an infinite sequence Z D .zn / and the associated operator DZ . 1) DZ is bounded iff .zn / is a bounded sequence. 2) DZ is self-adjoint iff zn 2 R for all n. 3) DZ is compact iff lim zn D 0. n!1 P 2 iff 1 4) DZ is Hilbert–Schmidt nD1 jzn j < 1. P1 5) DZ is trace class iff P nD1 jzn j < 1. p 6) For p  1, DZ 2 Jp iff 1 nD1 jzn j < 1. If .zn / is a finite sequence, then DZ is bounded, compact, and in Jp for each p  1. Proof Since kDn k D jzn j, for each n 2 N, we have kDZ k D supn jzn j. Then DZ is bounded Liff .zn / is a bounded sequence. For 2), consider the sequence of operators DN D NnD1 Dn , for N 2 N, as an operator on EZ by letting it act as multiplication by 0 on the remaining components. Thus DN is a finite rank operator on EZ for each N. Then kDZ  DN k D supk>N jzk j and so if limn!1 zn D 0, we then have that DZ is the norm limit of finite rank operators and thus is compact. On the other hand, if limn!1 zn ¤ 0, then fen g is a bounded sequence of vectors such that fDZ en g has no convergent subsequence. Thus, DZ is not compact. For 3) and 4), assume that DZ is compact. Then since DZ en D zn en and the fact that fen g forms an orthonormal basis for EZ we know the singular values of EZ are fzn g, which is just the sequence of numbers jzn j arranged in nonincreasing P1 order. Thus EZ is Hilbert–Schmidt iff P1 2 jz j < 1, and, trace class iff n nD1 nD1 jzn j < 1 and, more generally, in Jp iff P1 p nD1 jzn j < 1. Finally, if .zn / is a finite sequence, then DZ is actually a finite rank t operator and is trivially bounded, compact, and in Jp for each p  1. u

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Note that the above result is well known from the theory of multiplication operators on sequence spaces and is just translated here in our setting. Now that we have a formulation that can indeed give us a trace class operator, we can state the result we will use to fully recover certain functions of interest. P1 Corollary 2 If fzn g is a sequence Q of complex numbers satisfying nD1 jzn j < 1, then we have det.I  zDZ / D n .1  zn z/. Proof This is a direct consequence of Eq. (8) for trace class operators of which DZ is one when the series is absolutely summable. u t By the previous corollary, we can now see that we will be getting an entire function out of our construction. Thus if we want to handle meromorphic functions, we will need to handle zeros and poles separately. Also, we will want to choose our sequence .zn / to be the reciprocals of the zeros and, separately, of the poles. With this in mind, we make the following final construction for our operator associated with a meromorphic function. First, let f .z/ be an entire function on C with z D 0 not a zero of f . Let fan g be a sequence of the zeros of f .z/, counting multiplicity. Define the sequence Z D .zn / where zn D a1n . Define DZ as before and call this Df . Now given an integer m  1, if we have Df 2 Jm n Jm1 , then detm .I  zDZ / is well defined, where the regularized determinant detm was defined in Sect. 2. (See, especially, Definition 6 and Theorem 5.) Finally, we note that if we are dealing with a meromorphic function instead of an entire function, we follow the lead from the proof of the Weil conjectures to simply take the ratio of these regularized determinants, with possibly the operator associated with the numerator being in different trace classes (that is, in different operator ideals) than that of the denominator. In the next section, we will examine what this construction accomplishes for several classically important functions in number theory.

5 Applications of the Construction In this section, we apply our construction to some special meromorphic functions of interest and conclude with showing that this construction does indeed give a suitable replacement for the Frobenius for any entire function of finite order.

5.1 Rational Functions To begin, we look at the simplest type of meromorphic functions: the rational functions. Let f .z/ be a rational function. Then, we can write f .z/ D zk g.z/ for  s  Y h.z/ z some k 2 Z and further g.z/ D g.0/ , where h.z/ D 1 and k.z/ an nD1

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55

 t  Y z 1 for some finite set fa1 ; a2 ; : : : ; as ; b1 ; b2 ; : : : ; bt g. Construct bn nD1 the operator Dh.z/ and Dk.z/ as given in the previous chapter. The following theorem tells us that the determinant exactly recovers the given function f . k.z/ D

Theorem 20 If f .z/ is a rational function as above, then f .z/ D zk g.0/

det1 .I  zDh.z/ / : det1 .I  zDk.z/ /

(20)

Proof Write out f .z/ as given in the preceding paragraph. Then consider the finite sequences Z D f a11 ; a12 ; : : : ; a1s g and P D f b11 ; b12 ; : : : ; b1t g. The operators DZ and DP are both trivially trace class since both are created from finite sequences. Hence, we may apply the 1-regularized determinant (really, just the normal Fredholm determinant since both operators are trace class) we defined to obtain Qs z det1 .I  zDh.z/ / g.z/ nD1 .1  an / D D Qt (21) z det1 .I  zDk.z/ / g.0/ .1  / nD1 b n

det .IzD

/

and thus, f .z/ D zk g.0/ det11 .IzDh.z/ .u t k.z/ / We consider Dh.z/ to be the analog of Frobenius for h.z/ (zeros) whereas Dk.z/ would be the analog for k.z/ (poles). Then this ratio of determinants would be considered a graded determinant associated with the Frobenius of the divisor (zeros minus poles) of g.z/.

5.2 Zeta Functions of Curves over Finite Fields Recall that the zeta function of a (smooth, algebraic) curve Y over the finite field ! 1 X Yn ns q Fq is defined as Y .s/ D exp . The proof of the Weil conjectures n nD1 expressed this function as an alternating product of determinants as follows: Y .s/ D

det.I  F  qs jH 1 / : det.I  F  qs jH 0 / det.I  F  qs jH 2 /

One of the Weil conjectures, that Y .s/ is a rational function of qs , then followed from this formula. Thus we may apply the result in the previous section about rational functions to obtain the following theorem. Theorem 21 Let Y be a smooth, projective, geometrically connected curve over Fq , f .qs / the field with q elements. Write Y .s/ D g.q s / with f .z/; g.z/ both polynomials. Then   det1 I  qs Df  : Y .s/ D (22) det1 I  qs Dg

56

T. Cobler and M.L. Lapidus f .z/ is a g.z/ det1 .IzDf .z/ / det1 .IzDg.z/ /

Proof We have that

rational function of z. Thus by the rational function

f .z/ result: g.z/ det1 .Iqs Df /

and so replacing with z D qs gives: Y .s/ D

det1 .Iqs Dg /

D

, as desired. u t

Note that the results in this subsection can be extended in a straightforward manner to the zeta function of a (smooth, algebraic) d-dimensional variety over Fq , where the integer d  1 is arbitrary.

5.3 The Gamma Function The next meromorphic function that we Z 1will turn our attention to is the Gamma function, defined initially by  .z/ D xz1 ex dx for 1. This function 0

has numerous applications in many branches of mathematics, including our focus— number theory. One point of interest is that this function gives a meromorphic continuation to all of C of the factorial function on integers. It also appears in the functional equation for the Riemann zeta function. We have the following wellknown properties of the Gamma function: Theorem 22 For z 2 C, z … f0; 1; 2; : : :g, we have 1)  .z C 1/ D z .z/. 2)  .n/ D .n  1/Š for n 2 N.  1  ez Y  z 1 z n 3)  .z/ D 1C e . z nD1 n This infinite product representation for  .z/ allows us to now show that we can recover the function from the determinant of the operator construction we have laid out. Theorem 23 We have that for all z 2 C,  .z/ D

ez 1 : z det2 .I  zDz .z/ /

(23)

Proof We will apply our construction to the function g.z/ D z1.z/ . This function is entire and has a simple zero at each negative integer. Note that the residue of  .z/ at z D 0 is 1; so that g.0/ D 1. Now if we consider the sequence, Z D . 1n /, of reciprocals of zeros of g.z/, we see that it is not a summable series but that it is square summable. This means the associated operator DZ is not trace class, but only Hilbert–Schmidt. This forces us to use det2 in our definition of the regularized determinant. In fact, det2 .I  zDZ / D

1 h Y z z i e n : 1C n nD1

(24)

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57

This then leads to the following computation: 1 det2 .IzDz .z/ /

D det2 .I  zDZ /1 1 h Y

D

nD1 1  Y

D

!1 z z i n 1C e n

1C

nD1

z 1 z en n



D zez  .z/: Therefore, we conclude that  .z/ is given by Eq. (23), as desired. u t

5.4 The Riemann Zeta Function Next, we turn our attention to another important example, the Riemann zeta function. First, we will consider the well-known Euler product expression for .s/. Theorem 24 For s 2 C, with 1, .s/ D

Y

.1  ps /1 ;

p

where the product is taken over all prime numbers p. To use our formulation, let .z/ D 1z. Then, by the result for rational functions in Sect. 5.1 (Theorem 20), we have that .z/ D det.I  zD /, which is true for every value of z ¤ 1. Letting z D ps then gives .1  ps /1 D .det.I  ps D //1 for s ¤ 2ik , k 2 Z. This leads to the following operator based Euler product: log p Theorem 25 For s 2 C, with 1, .s/ D

Y

.det.I  ps D //1 ;

(25)

p

where the product is taken over the primes p. Proof We simply apply the determinant equality to each term in the infinite product and then use the standard Euler product convergence. Note that for 1, we never have s D 2ik for any integer k; so that the determinant equality does apply at log p each prime p. u t   s The completed zeta function, .s/ D 12   2 s.s  1/ 2s .s/, is an entire function whose zeros all lie in the critical strip fs 2 C W 0 < 0 such that jf .z/j < exp.jzja / for jzj > r0 . If f is not of finite order, then f is said to be of infinite order. If f is of finite order, then the number D inffa W jf .z/j < exp.jzja / for jzj sufficiently largeg is called the order of f . Thus the order of an entire function f is a measure of the growth of jf .z/j as jzj ! 1 whereas the rank of f is based on the growth of the nth smallest zero as n ! 1. From the definitions, there is no inherent relationship between the two concepts, but with the following version of the Hadamard factorization theorem, we see that they are in fact closely related:

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Theorem 30 (Hadamard Factorization Theorem) If f .z/ is an entire function of finite order , then f has finite genus   and f admits the following factorization:   1 Y z m g.z/ f .z/ D z e ; (29) Ep an nD1 where g.z/ is a polynomial of degree q  and p D Π, the integer part of . In particular, f is of rank not exceeding p. Now when we apply our operator construction to a given entire function of finite order we obtain a Quantized Hadamard Factorization Theorem. Theorem 31 (Quantized Hadamard Factorization Theorem) If f .z/ is an entire function of finite order , then f admits the following factorization: f .z/ D zm eg.z/ detpC1 .I  zDf .z/ /;

(30)

where g.z/ is a polynomial of degree q  , and p D Π. Proof By the standard Hadamard factorization theorem, we can write f .z/ D zm eg.z/

1 Y

 Ep

nD1

z an

 ;

(31)

where g.z/ is a polynomial of degree q  and p D Π, with the rank of f zeros of f .z/ including not exceeding p. That P is, if1 fa1 ; a2 ; : : :g is the multiset of 1 1 multiplicity, then 1 < 1. Thus if Z D f ; ; : : :g, the associated nD1 jan jpC1 a1 a2 operator DZ 2 JpC1 . Then we can calculate successively: detpC1 .I  zDf .z/ / D detpC1 .I  zDZ / 0 2 13   p 1 j Y X z z 4 1 A5 D exp @ j a n ja n nD1 jD1 D

1 Y

 Ep

nD1

D

f .z/ zm eg.z/

z an



:

t Thus we have that f .z/ D zm eg.z/ detpC1 .I  zDf .z/ /, as desired. u In the above proof, we see that the extra convergence factor in the regularized determinants is exactly the same as the one for the elementary factor in the infinite product representation of entire functions, which validates, in some sense, the choice in this paper for the type of regularized determinants as those based on trace ideals. Thus the convergence factors needed in the usual Hadamard Factorization Theorem have an interpretation here relating to Tr.Dn /.

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6 Conclusion We ended the previous section by giving what we called the Quantized Hadamard Factorization Theorem. This showed that the construction given in this paper can apply to any entire function of finite order and then, by taking ratios of determinants, can be extended to any meromorphic function which is a ratio of two such entire functions. This was worked out explicitly for the Riemann zeta function (see Theorems 27 and 28 above), and it has also been worked out by the authors for zeta functions of self-similar strings, both in the lattice and nonlattice case. (See [28, Chaps. 2 and 3] for background on self-similar fractal strings.) However, there was nothing in the construction preventing us from applying our results to even more general number-theoretic functions. In particular, a natural idea would be to try to extend our determinant formulas to other L-functions (see [33]). Could we then apply this construction to any zeta function (or, at least, to most zeta functions) from arithmetic geometry? This would require, essentially, knowledge about the existence of suitable meromorphic extensions of such functions, as well as about the asymptotic behavior of the zeros and poles of such extensions. Phrased differently, the L-functions for which our methods could be applied are those which can be suitably completed to become entire functions of finite order (or ratios of such entire functions). Furthermore, this naturally brings the consideration of the Selberg class of functions. See [33] or [25, Appendix E] (and the many references therein) for a discussion of these functions. Another direction to take is to further justify why using ratios of these determinants is the correct method for handling meromorphic functions. In [25], the second author considers the properties of the Riemann zeta function as related to supersymmetric theory in physics and this ratio of determinants can be explained as a (regularized) Berezinian determinant from the theory of super linear algebra. However, this will not be further discussed here, but could be crucial for expanding upon the ideas presented in this work. In this paper, we obtained a quantized version of the Hadamard factorization theorem, Theorem 31, but we expect to be able to generalize this result to obtain a quantized Weierstrass product formula; see [5]. With all of the successes obtained here, we must also admit the failures of this theory, at least in its present stage of its development. The construction of the operator for .s/ explicitly assumed knowledge of the zeros of .s/ and thus one could never hope to prove (RH) directly with this method. However, if we could find a different way to obtain the same function, by comparison you could extrapolate the zeros as was done with the Weil conjectures. That is, we need a suitable geometry and cohomology theory that would result in the same ratios of determinants of these operators. In the Weil conjectures, the geometry or points on the curve (over Fq , the algebraic closure of Fq ) corresponded to the fixed points of powers (or iterates) of the Frobenius operator. (Recall from our discussion just prior to Sect. 4.1 that in our context, the “fractal cohomology space” would seem to be the total eigenspace EZ to which we restricted the original generalized Polya–Hilbert operator, viewed

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as Frobenius acting on an appropriate analog of the underlying “curve.”) Could we then consider the fixed points of the operator constructed in this paper? Analytically, this can be done by considering a suitable notion of generalized eigenfunctions (viewed as generalized tempered distributions). Thus far, however, this idea has not led to the development of a suitable working theory for the geometry underlying .s/. Providing an appropriate geometric framework is one of our main long-term objectives for future research on this subject. Another interesting and related question (connected, in particular, to our discussion in Sects. 1.1 and 1.3) is whether the still conjectural fractal cohomology theory satisfies a suitable analogue of the Lefschetz fixed point formula (as stated in Theorem 1) for the counterpart of Frobenius. One additional plan that we are currently working on is to rephrase the construction we have described here as a cohomology of sheaves in order to properly transition from the local setup given in this paper to a more global approach that might give new and useful information. Acknowledgements The work of Michel L. Lapidus was supported by the US National Science Foundation (NSF) under the grant DMS-1107750.

References 1. E. Artin, Quadratische Körper im gebiet der höheren Kongruenzen, I and II. Math. Zeitschr. 19, 153–206, 207–246 (1924) 2. A. Atzmon, B. Brive, Surjectivity and invariant subspaces of differential operators on weighted Bergman spaces of entire functions, in Contemporary Mathematics, ed. by A. Borichev, H. Hedenmalm, K. Zhu, vol. 404 (American Mathematical Society, Providence, 2006), pp. 27–39 3. M.V. Berry, The Bakerian lecture: quantum chaology. Proc. R. Soc. A 413, 183–198 (1987) 4. M.V. Berry, J.P. Keating, The Riemann zeros and eigenvalue asymptotics. SIAM Rev. 41, 236–266 (1999) 5. T. Cobler, M.L. Lapidus, Zeta functions and Weierstrass’ factorization theorem via regularized determinants and infinitesimal shifts in weighted Bergman spaces (in preparation, 2017) 6. A. Connes, Trace formula in noncommutative geometry and the zeros of the Riemann zeta function. Sel. Math., New Ser. 5, 29–106 (1999) 7. J.B. Conway, Functions of One Complex Variable. Graduate Texts in Mathematics, vol. 11, 2nd ed. (Springer, New York, 1995) 8. P. Deligne, La conjecture de Weil: I. Publ. Math. l’IHÉS 43, 273–307 (1974) 9. P. Deligne, La conjecture de Weil: II. Publ. Math. l’IHÉS 52, 137–252 (1980) 10. C. Deninger, Evidence for a cohomological approach to analytic number theory, in First European Congress of Mathematics, ed. by A. Joseph, F. Mignot, F. Murat, B. Prum, R. Rentschler. Progress in Mathematics, vol. 3 (Birkhäuser, Basel, 1994), pp. 491–510 11. C. Deninger, Some analogies between number theory and dynamical systems on foliated spaces, in Proceedings of International Congress of Mathematicians (Berlin, 1998), vol. I (1998), pp. 163–186. Documenta Math. J. DMV (Extra Volume ICM 98) 12. J. Dieudonné, On the history of the Weil conjectures. Math. Intell. 10, 7–21 (1975) 13. H.M. Edwards, Riemann’s Zeta Function, Dover edn. (Dover Publications, Mineola, 2001)

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14. A. Grothendieck, The cohomology theory of abstract algebraic varieties, in Proceedings of International Congress of Mathematicians (Edinburgh, 1958) (Cambridge University Press, New York, 1960), pp. 103–118 15. A. Grothendieck, Formule de Lefschetz et rationalité des fonctions l. Séminaire N. Bourbaki Exp. 279, 41–55 (1964/1966) 16. A. Grothendieck, Standard conjectures on algebraic cycles, in Algebraic Geometry (Internationlal Colloquium, Tata Institute of Fundamental Research, Bombay, 1968) (1969), pp. 193–199 17. S. Haran, The Mysteries of the Real Prime. London Mathematical Society Monographs, New Series, vol. 25 (Oxford Science Publications, Oxford University Press, Oxford, 2001) 18. H. Hasse, Abstrakte Begründung der komplexen Multiplikation und Riemannsche Vermutung in Funktionenkörpern. Anh. Math. Sem. Hamburg 10, 325–348 (1934) 19. H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman Spaces. Graduate Texts in Mathematics, vol. 199 (Springer, New York, 2000) 20. H. Herichi, M.L. Lapidus, Riemann zeros and phase transitions via the spectral operator on fractal strings. J. Phys. A Math. Theor. 45, 374005 (23 pp.) (2012) 21. H. Herichi, M.L. Lapidus, Fractal Complex Dimensions, Riemann Hypothesis and Invertibility of the Spectral Operator. Contemporary Mathematics, vol. 600 (American Mathematical Society, Providence, 2013), pp. 51–89 22. H. Herichi, M.L. Lapidus, Truncated infinitesimal shifts, spectral operators and quantized universality of the Riemann zeta function. Ann. Fac. Sci. Toulouse Math. 23(3), 621–664 (2014) [Special issue dedicated to Christophe Soulé on the occasion of his 60th birthday] 23. H. Herichi, M.L. Lapidus, Quantized Number Theory, Fractal Strings and the Riemann Hypothesis: From Spectral Operators to Phase Transitions and Universality. Research Monograph (World Scientific, Singapore and London, 2017) approx. 360 pp. 24. N. Katz, An overview of Deligne’s proof of the Riemann hypothesis for varieties over finite fields, in Proceedings of Symposia Pure Mathematics, vol. 28 (American Mathematical Society, Providence, 1976), pp. 275–305 25. M.L. Lapidus, In Search of the Riemann Zeros: Strings, Fractal Membranes and Noncommutative Spacetimes (American Mathematical Society, Providence, 2008) 26. M.L. Lapidus, Towards quantized number theory: spectral operators and an asymmetric criterion for the Riemann hypothesis. Philos. Trans. R. Soc. A 373(2047), 24 pp. (2015) 27. M.L. Lapidus, H. Maier, The Riemann hypothesis and inverse spectral problems for fractal strings. J. Lond. Math. Soc. 52(1), 15–34 (1995) 28. M.L. Lapidus, M. van Frankenhuijsen, Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings. Springer Monographs in Mathematics (Springer, New York, 2013). Second revised and enlarged edition of the 2006 edition 29. F. Oort, The Weil conjectures. Nw. Archief v. Wiskunde 5 Ser. 15(3), 211–219 (2014) 30. S.J. Patterson, An Introduction to the Theory of the Riemann Zeta-Function (Cambridge University Press, Cambridge, 1988) 31. B. Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsb. der Berliner Akad. 671–680 (1858/1860) 32. W. Rudin, Functional Analysis, 2nd edn. (McGraw Hill, New York, 1991) 33. P. Sarnak, L-functions, in Proceedings of the International Congress of Mathematicians, vol. I, Berlin (1998), pp. 453–465. Documenta Math. J. DMV (Extra Volume ICM 98) 34. F.K. Schmidt, Analytische Zahlentheorie in Körpern der Charakteristik p. Math. Zeitschr. 33, 1–32 (1931) 35. B. Simon, Notes on infinite determinants of Hilbert space operators. Adv. Math. 24, 244–273 (1977) 36. B. Simon, Trace Ideals and Their Applications. Mathematical Surveys and Monographs, vol. 120, 2nd edn. (American Mathematical Society, Providence, 2005) 37. E.C. Titchmarsh, The Theory of the Riemann Zeta Function. Oxford Mathematical Monographs, 2nd edn. (Oxford University Press, Oxford, 1986) (revised by D.R. Heath-Brown)

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The Temptation of the Exceptional Characters John B. Friedlander and Henryk Iwaniec

In celebration of forty years of collaboration

Abstract We survey some of the history and results related to the topic of the title with an emphasis admittedly biased toward our joint works thereon.

1 Introduction Both on our own and together, the authors have been fascinated by the problems posed by the possible existence of exceptional Dirichlet characters and the profound influence these would play in some of the basic questions of multiplicative number theory. We take this opportunity to survey some of the history and results related to this topic, with an emphasis admittedly biased toward our joint works thereon.

2 The Class Number Formula Let K be a number field and let D denote the absolute value of its discriminant. Recall that the Dedekind zeta-function of K X  C OK .1/ .Na/s D K .s/ D s  1 a

J.B. Friedlander () Department of Mathematics, University of Toronto, Toronto, ON, Canada M5S 2E4 e-mail: [email protected] H. Iwaniec Department of Mathematics, Rutgers University Piscataway, New Brunswick, NJ 08903, USA e-mail: [email protected] © Springer International Publishing AG 2017 H. Montgomery et al. (eds.), Exploring the Riemann Zeta Function, DOI 10.1007/978-3-319-59969-4_4

67

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has a simple pole at s D 1, with residue  given in terms of some of the most important field constants by D

2r1 Cr2  r2 h.K/R p ! D

where h is the (wide) class number, R is the regulator, ! is the number of roots of unity, and r1 ; r2 are the number of real (respectively, complex conjugate pairs of) embeddings of K into the complex plane. p We shall throughout restrict ourselves to quadratic fields, so K D Q. .1/D/ where D D is the real primitive Dirichlet character of conductor D. Now, the above constants simplify substantially and, by the Dirichlet class number formula, the residue is also given by X L.1; / D

.n/n1 : n>1

We shall now denote the class number by h.˙D/ rather than h.K/ and, since the class number is the order of a finite group, it follows that L.1; D / > 0, which we recall was a pivotal step in Dirichlet’s proof [4]: Theorem 1 Let q be a positive integer and a an integer relatively prime to q. Then there are infinitely many prime numbers p with p  a .modq/ :

3 The Size of L.1; D / It is important to have as much information as we can about the size of Dirichlet L-functions in general, especially at s D 1 and most especially for D D . As far as upper bounds, we know that L.1; / log D and that, under assumption of the Riemann Hypothesis for the L-function in question, we have L.1; / log log D : One would like to know more but the above is pretty good when compared to the situation for lower bounds where one knows, under the same RH, that L.1; /  .log log D/1

Exceptional Characters

69

but what we know unconditionallypis much less. From the class number formula it follows quickly that L.1; /  1= D. Moreover, for even characters one improves this by a factor of log D by virtue of the presence of a non-trivial regulator.

4 The Zero-Free Region It is important to have a good zero-free region for L.s; /, s D C it, for all , but the real characters give extra problems (closely related to the size of L.1; /). Specifically: Theorem 2 (de la Vallée-Poussin, Landau [16]) There exists a constant c > 0 such that L.s; / ¤ 0 for > 1 

c ; log.D.jtj C 1//

with the possible exception of a single real and simple zero ˇ D ˇD . We call such D; ; ˇ “exceptional (with respect to the constant c)” and Landau proved that they are just that! More precisely: Theorem 3 (Landau [16]) For any given positive M there exists c D c.M/ > 0 such that any given sequence of conductors Dn exceptional with respect to the constant c satisfies DnC1 > DM n for all n. Note that, in order for there to be any exceptional conductors, there must be an infinite family of them (since otherwise, knowing as we do that L.1; / > 0, we could just choose c sufficiently small so as to rectify the finite number of obstructions to the inequality).

5 The Class Number Theorem 4 (Hecke, Landau [16]) As D runs through non-exceptional conductors, we have L.1; / 

1 log D

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and hence, for imaginary quadratic fields p D : h.D/  log D A breakthrough idea came from Deuring [3]. Theorem 5 (Deuring [3], Mordell [19]) If the Riemann zeta-function has any nontrivial zero off the line D 12 then, for imaginary quadratic fields h.D/ ! 1 as D ! 1: Deuring’s idea was extended by Heilbronn [15] to give: Theorem 6 (Heilbronn) If any Dirichlet L-function has any non-trivial zero off the line D 12 then, for imaginary quadratic fields h.D/ ! 1 as D ! 1: Combining this last theorem with Hecke’s result, [15] proves a famous conjecture of Gauss: Corollary 1 (Heilbronn) For imaginary quadratic fields h.D/ ! 1 as D ! 1: Improving the quantitative estimate in Heilbronn’s argument, Chowla became the first to deduce Theorem 7 (Chowla [2]) There are only a finite number of Euler’s idoneal numbers (discriminants for which there is only one class in each genus). Indeed, he shows that the number of classes in the principal genus goes to infinity. The first to beat the trivial class number bound by a fixed power was (again) Landau. Theorem 8 (Landau [17]) We have 1

3

h.D/  D 8 =.log D/ 4 : Siegel sharpened that result to give what we know today, namely: Theorem 9 (Siegel [20]) For every " and some c" > 0, we have L.1; / > c" D" : Corollary 2 We have 1

h.D/  D 2 " :

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71

Although, Siegel’s paper does not mention zeros of L- functions, the theorem incidentally returns attention from zeros off the line to zeros near s D 1 since, by the mean-value theorem of differential calculus, one has the following result. Corollary 3 For every " and some c" > 0, we have L. ; / ¤ 0 for

> 1  c" D" :

6 Primes in Arithmetic Progressions Shortly after the proofs of the prime number theorem by Hadamard and de la Vallée-Poussin, the latter combined his techniques with those of Dirichlet to give an asymptotic formula for the number of primes up to a given point, in an arithmetic progression. We take q>1, .a; q/ D 1, and then have .xI q; a/ D

X

1

px pa. mod q/

.x/ '.q/

and are interested in the range of uniformity in q for which this asymptotic formula can be shown to hold. There are three ranges for which this question is particularly noteworthy: q " x1" ; which is expected and would be a dream come true; 1

q " x 2 " ; which follows from the assumption of RH for all L-functions with characters of modulus q: q N .log x/N which is known to hold for arbitrary N (the Siegel–Walfisz theorem). An important related question is that of giving good upper bounds for p` D p` .q; a/, the least prime in the arithmetic progression in question. As immediate consequence of the above three ranges for the asymptotic formula one has p` q1C" ;

p` q2C" ;

p` exp.q1=N / :

It was an achievement of the highest order when Linnik [18] gave an unconditional proof of a bound with a fixed power.

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Theorem 10 Let q>2 and .a; q/ D 1. There exists a positive universal constant L such that p` .q; a/ qL : Many improvements have ensued in the known admissible values of “Linnik’s constant” L. It seems inevitable that separate proofs are required for the two different cases, depending on the existence or non-existence of exceptional zeros/characters. At first it seemed that the arguments in the exceptional case were the more difficult but more recently it has emerged that sometimes they can be simpler and stronger. Thus, there is the following theorem of Heath-Brown [14]. Theorem 11 Let " > 0. Then, there exists c."/ such that if there is a zero ˇ>1  c."/= log q then p` .q; a/ q2C" : What about other problems?

7 Twin Primes Linnik’s theorem offers an important example wherein the complete result can be proven, that is with or without the assumption of exceptional characters. It turns out, however, that there are very striking statements which can (up to the present) be proven only under the assumption of exceptional characters while proofs in the non-exceptional case remain a (probably remote) goal. An example of this phenomenon was given by the following surprising theorem of Heath-Brown [13]. Theorem 12 One (at least) of the following two alternatives must hold. 1) There exists a positive constant c such that, for every q and every mod q there are no real zeros ˇ of L.s; / with ˇ >1

c ; log q

or 2) there are infinitely many prime pairs p; p C 2. Remark 1 Recently, in [12] we have given a rather different account of this theorem.

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8 More Primes in Arithmetic Progressions For the remainder of the paper we are going to survey some of our joint works on the consequences of the assumption of the existence of exceptional characters. Here is a special case of our first work on this topic [7]. Theorem 13 Let .a; q/ D 1, r D 554; 401, and D D . For,   r 1 ; Dr 6q6 exp L.1; /.r C1/ we have p` .q; a/ q21=59 : Actually, we get the asymptotic formula in this range. It is remarkable that one gets below the GRH exponent 2, but there are two important cautions. 1. The result only holds for intervals of q restricted in terms of D. 2. The interval is non-empty only under the “exceptional” assumption that r

L.1; D / .log D/r 1 :

9 Primes in Short Intervals In [8] we proved the following result which, for short intervals, breaks the analogous Riemann Hypothesis barrier, albeit under constraints similar to those in the previous section. Theorem 14 Let D D , x>Dr with r D 18; 290 and 1

1

x 2  158 < y6x : Then Z

x

.x/  .x  y/ D xy

o  dt n r 1 C O L.1; /.log x/r : log t

Corollary 4 If r

L.1; / .log x/r 1 then Z

x

.x/  .x  y/ D xy

  1 o dt n : 1CO log t log x

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It may seem strange to see the assumed upper bound for L.1; / phrased in terms of log x rather than log D but, of course, these are of the same order of magnitude when x is about a fixed power of D.

10 Prime Values of Polynomials Recall that unconditionally we know [6] that there are infinitely many primes of the form a2 C b4 . We hoped but did not succeed to prove that under the assumption of exceptional characters one might deduce an infinitude of primes of the form a2 C 1. With a lot more work than in our other theorems of this type, we made a little progress. We state a couple of consequences of the results in [9]. R1p Corollary 5 Let  D 0 1  t6 dt. Suppose that L.1; D /6.log D/200 : Then, for     exp .log D/10 < x < exp .log D/16 we have X a2 Cb6 6x

  n   1 o 4  a2 C b6 D x2=3 1 C O  log x

so, if there are an infinity of such D, then there are an infinity of primes of the form a2 C b6 . Under a very similar assumption, one deduces the infinitude of elliptic curves over the rationals having prime discriminants, hence only a single prime of bad reduction.

11 Squares Plus Primes Let D ! 1 through fundamental discriminants (so, real quadratic fields). Let N .D/ denote the number of solutions of D D m2 C p : We write .p/ D 1 C D .p/ and note that this is just the number of solutions of the congruence b2  D.modp/.

Exceptional Characters

75

p One conjectures that N .D/  H D= log D where   Y 1 1 .p/  1 1 : HD p p p Let us define V.D/ D

 Y .p/ 1 p p 0 and `.D/ < 1, then    p  p 2 N .D/ D DV.D/ 1 C O exp  log 1=`.D/ : 5 Thus, we see: Corollary 6 i) If `.D/ is less than a sufficiently small positive constant c, then p N DV.D/ ii) If `.D/ ! 0 as D ! 1, then N 

p DV.D/ :

Note that by Mertens’ theorem we have V.D/  e H= log D, so the last corollary is presumably giving us an answer that is wrong by the factor e . This is reminiscent of the wrong factor spit out by the sieve of Eratosthenes in a first attempt at counting primes (and for essentially the same reason).

12 Some Ideas About the Proofs For a glimpse of the main idea involved one can argue as follows. Let be real and let us take s D 1 C " so that we have the Euler product Y 1 1  .p/ps L.s; / D p

and log L.s; / D

X p

.p/ps C O" .1/ :

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Now, L.s; / is small if and only if log L.s; / is large and negative and this can happen only if we have .p/ D 1 for most small primes p. Thus, on squarefree integers should behave like the Möbius function . This reminds us of the sieve. But is periodic and, if we are looking near x which is huge compared to D, then the period is small, almost fixed, so our “sieve” is supercharged with very special (indeed, illusory) properties. On one hand, the character’s periodicity enables harmonic analysis, on the other hand, its Möbius-like feature detects primes. The combination of these features brings tastes of the Hilbert–Polya dream. In the next two sections, we shall sketch two different ways this simple idea can be utilized.

13 The Exceptional Conductor Influencing Itself In this section we sketch the proof of the last stated theorem, that is we consider the equation D D m2 C p. We begin with the finite sequence A W n D D  m2

01

Note that .d/ D 0 if there is any prime p j d with .p/ D 1 so that, if is exceptional then usually .d/ D .d/ D 0.

Exceptional Characters

77

We next recall that by the fundamental lemma of sieve theory, we have X ˚  S.A ; z/ D xV.z/ 1 C O.es / C .d/ : d6y

Here, the parameter y may be chosen arbitrarily subject to y>z, j j61 ;

sD

log y log z

and

V.z/ D

Y

1

pD

!

X

0 .d/.m/ :

dmDn

Here,  is easy to handle since the problematic factor , which hides the Möbius function, occurs with small m and 0 behaves like . On the other hand,  contributes to the error term in what appears at first glance to be unmanageable form. Specifically, if we use the simple bound j .n/j6

X

0 .d/ .m/ ;

dmDn

then this behaves like the divisor function  2 or 4 and, for almost any sequence one can name, we have pretty limited knowledge of its distribution. One more elementary reduction is needed, a special case of Corollary 22.11 of [10].

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Lemma 1 Let r>2,  D x1=r , d6x. Then X A 2.ı/  .d/6 ıjd ı6

with some A r log r. Inputting the lemma, we see that, for n  x, j .n/j6.log x/

X A X 2.ı/ .m/ : ı6

ı`mDn

Now, for each fixed ı we have replaced 0 by 1 which is a function of type 3 . In this way, the problemP is reduced (of course, modulo technical details) to having to require knowledge of n6x an k .n/ only for k D 1; 2; 3. We have sufficient knowledge of these arithmetic functions to sometimes go beyond the square-root barrier, that is q > x1=2C in the case of arithmetic progressions due to [5], and y < x1=2 in the case of short intervals, already much earlier due to Atkinson [1]. One may say that the crossing of this barrier is due to the power of harmonic analysis on GL.3/.

References 1. F.V. Atkinson, A divisor problem. Quart. J. Math. (Oxford) 12, 193–200 (1941) 2. S. Chowla, An extension of Heilbronn’s class-number theorem. Quart. J. Math. (Oxford) 5, 304–307 (1934) 3. M. Deuring, Imaginäre quadratische Zahlkörper mit der Klassenzahl 1. Math. Z. 37, 405–415 (1933) 4. G.L. Dirichlet, Beweiss des Satzes das jede arithmetische Progression deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält, Werke, Kön. Preuss. Akad. der Wissen. (Berlin, 1889), pp. 313–342 (Reprinted Chelsea, New York, 1969) 5. J.B. Friedlander, H. Iwaniec, Incomplete Kloosterman sums and a divisor problem, with an appendix by B.J. Birch and E. Bombieri. Ann. Math. 121, 319–350 (1985) 6. J.B. Friedlander, H. Iwaniec, The polynomial X 2 C Y 4 captures its primes. Ann. Math. 148, 945–1040 (1998) 7. J.B. Friedlander, H. Iwaniec, Exceptional characters and prime numbers in arithmetic progressions. Int. Math. Res. Not. 37, 2033–2050 (2003) 8. J.B. Friedlander, H. Iwaniec, Exceptional zeros and prime numbers in short intervals. Sel. Math. N. Ser. 10, 61–69 (2004) 9. J.B. Friedlander, H. Iwaniec, The illusory sieve. Int. J. Number Theory 1, 459–494 (2005) 10. J.B. Friedlander, H. Iwaniec, Opera de Cribro, Colloquium Publications, vol. 57 (American Mathematical Society, Providence, RI, 2010), xx + 527 pp. 11. J.B. Friedlander, H. Iwaniec, Exceptional discriminants are the sum of a square and a prime. Quart. J. Math. (Oxford) 64, 1099–1107 (2013) 12. J.B. Friedlander, H. Iwaniec, Twin primes via exceptional characters. arXiv:1607.03261 13. D.R. Heath-Brown, Prime twins and Siegel zeros. Proc. Lond. Math. Soc. 47, 193–224 (1983)

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14. D.R. Heath-Brown, Siegel zeros and the least prime in an arithmetic progression. Quart. J. Math. (Oxford) 41, 405–414 (1990) 15. H.A. Heilbronn, On the class-number in imaginary quadratic fields. Quart. J. Math. (Oxford) 5, 150–160 (1934) 16. E. Landau, Über die Klassenzahl imaginär quadratischer Zahlkörper. Kön. Akad. Wiss. Gött. Nachr. 1918, 285–295 (1918) 17. E. Landau, Bemerkungen der Heilbronnschen Satz. Acta Arith. 1, 1–18 (1936) 18. Yu.V. Linnik, On the least prime in an arithmetic progression. II, The Deuring-Heilbronn phenomenon. Mat. Sbornik 57, 347–368 (1944) 19. L.J. Mordell, On the Riemann hypothesis and imaginary quadratic fields with a given class number. J. Lond. Math. Soc. 9, 405–415 (1934) 20. C.L. Siegel, Über die Classenzahl quadratischer Zahlkörper. Acta Arith. 1, 83–86 (1936)

Arthur’s Truncated Eisenstein Series for SL.2 ; Z/ and the Riemann Zeta Function: A Survey Dorian Goldfeld Abstract Eisenstein series are not in L2 . Maass, and later Selberg, naively truncated the constant term of Eisenstein series (for symmetric spaces of rank 1) so that the resulting truncated Eisenstein series was square integrable. This procedure was generalized to Eisenstein series on higher rank groups by Langlands and Arthur. In this survey we focus on the deep connections between Eisenstein series for SL.2; Z/, truncation, and the Riemann zeta function. Applications to zero free regions for the Riemann zeta function and automorphic L-functions are elucidated.

1 Introduction Let h WD fx C iy j x 2 R; y > 0g denote the upper half plane. Then, as is well known, the modular group  D SL.2; Z/ acts on h as follows. For  D ac db 2  and z 2 h, the action of  on z is given by  z WD .az C b/=.cz C d/: The Hilbert space L2 .nh/ of smooth L2 automorphic functions f W h ! C satisfying  f

az C b cz C d

 D f .z/;

  ab for all z 2 h; 2 ; cd

with Petersson inner product “ hf ; gi WD

f .z/  g.z/ nh

dxdy ; y2

f ; g 2 L2 .nh/ ;

has been extensively studied. Here f 2 L2 means Z

jf .x C iy/j2 nh

dxdy < 1; y2

D. Goldfeld () Mathematics Department, Columbia University, New York, NY 1002, USA e-mail: [email protected] © Springer International Publishing AG 2017 H. Montgomery et al. (eds.), Exploring the Riemann Zeta Function, DOI 10.1007/978-3-319-59969-4_5

83

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D. Goldfeld

where nh can be taken as the region F WD fz 2 h j jzj  1; j 0: The analytic continuation of the Riemann zeta function then gives the meromorphic continuation of the Eisenstein series to the whole complex plane. The modified Riemann hypothesis for a function Z.s/ asserts that all its zeros are either on the line 0, we have E.z; s/  ys C .s/y1s : Consequently Z

ˇ s ˇ ˇy C .s/y1s ˇ2 dxdy > y2 nh

Z

1 2

xD 12

Z

1

ˇ s ˇ ˇy C .s/y1s ˇ2 dxdy D 1 y2 yD0

for any value of s 2 C: It follows that the Eisenstein series is not in L2 .nh/: To overcome the fact that E.z; s/ is not in L2 and facilitate the computation of integrals involving Eisenstein series, Maass [16], in 1949, introduced the naive truncated Eisenstein series ET .z; s/. Let T > 0: Then for z 2 h, s 2 C, ( E.z; s/ if y < T; ET .z; s/ WD s s if y > T: E.z; s/  y  .s/y This function is not continuous in z but will lie in L2 : It is also not automorphic, i.e.,   az C b ; s ¤ ET .z; s/ ET cz C d   ab for most 2 . cd In 1962, Selberg [19] extended Maass’ work on naive truncated Eisenstein series to the more general situation of symmetric spaces of rank 1. The inner product of two truncated Eisenstein series (Maass–Selberg relations), which was based on Green’s theorem, played a crucial role in the computation of the Selberg trace formula [18]. The more general Arthur–Selberg trace formula (see [15]) has been central for the advancement of modern number theory. One of the most basic tools in the theory of automorphic forms on higher rank groups is the truncation operator which was first introduced by Langlands [14] in 1966. In 1978–1980 Arthur [1, 2] made an important advance by extending and making much more explicit Langlands’ truncation operator, especially when applied to Eisenstein series. In particular, Arthur’s truncation involves a summation P codim.F/ .1/ EF for a certain characteristic function EF over the faces F of a F certain convex polyhedron in a Euclidean space V.

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The key property we want to point out for this survey is that Arthur’s truncated Eisenstein series are automorphic which was not the case with Maass and Selberg’s naively truncated Eisenstein series. Arthur’s truncated Eisenstein series greatly simplified the computation of the Maass–Selberg relations, especially in the situation of higher rank groups. The use of differential operators (via Green’s theorem) was avoided and all proofs of the Maass–Selberg relations were reduced to algebraic computations as in the Rankin–Selberg method [9]. In this survey we focus on Arthur’s truncated Eisenstein series for the modular group  and its connections with the Riemann zeta function. The reader may find it useful to concurrently look at other surveys on truncation such as [4–6].

2 Arthur’s Truncated Eisenstein Series In order to define Arthur’s truncated Eisenstein series for  we require some preliminary definitions. A function F W h ! C is said to be periodic if F.z C 1/ D F.z/ for all z 2 h: Definition (Constant Term) For a piecewise continuous periodic function F W h ! C define Z 1 CF.z/ WD F.z C u/ du; .for z 2 h/; 0

to be the constant term of F: Definition (Truncated Constant Term) Let T > 0 and z D x C iy 2 h. For an integrable periodic function F W h ! C define ( 0 if 0  y  T; T C F .z/ WD CF.z/ if y > T; to be the truncated constant term of F. Definition (Arthur’s Truncation Operator) Let T > 0 and z 2 h. For an integrable periodic function F W h ! C define X ƒT F.z/ WD F.z/  CTF . z/; 21 n

to be the truncation of the function F.z/ with respect to T. Here 1 WD

  1m ; 0 1

is the stabilizer of 1 under the action of .

m2Z

Arthur’s Truncated Eisenstein Series for SL.2; Z/ and the Riemann Zeta. . .

87

Definition (Polynomial Growth) A periodic function  W h ! C has polynomial growth if there exist constants C; B > 0 such that j.z/j  C Im.z/B ;

.Im.z/ ! 1/:

The following theorem summarizes the main properties of the truncation operator ƒT . Theorem 2.1 (Properties of ƒT ) Let T > 0: The truncation operator ƒT satisfies the following properties. • Let  2 L2 .nh/ be a cusp form (the constant term of  is zero) then ƒT  D : • Let  W nh ! C be integrable. Assume T  1 and z 2 F: Then ƒT .z/ D   CT .z/: • Assume  W nh ! C is smooth, is an eigenfunction of the invariant Laplacian, and has polynomial growth. Then ƒT  2 L2 .nh/: • Let ;  2 L2 .nh/ : Then hƒT ; i D h; ƒT i: Proof of Theorem 2.1 It is clear that if  is a cusp form then truncation has no effect since the constant term of  is zero. Next, assume z lies in the fundamental domain F: Then for  2 =1 ; with  62 1 , the complex number  z must lie outside of F and not in a translate m C F with m 2 Z: It is easy to see that in this situation, we must have Im. z/ < 1: The second bulleted item immediately follows from this. The third bulleted item is much more difficult to prove, so we do not give the details. The last bulleted item is t u easily proved with a change of variables z 7!  1 z for  2 =1 : We now carefully examine the action of the truncation operator on the Eisenstein series E.z; s/ with Fourier expansion given in (1). Then CE.z; s/ D ys C .s/y1s ; and for T > 0;   CT E.z; s/ D ys C .s/y1s  charŒT; 1 .y/; where ( charŒ˛; ˇ .y/ D

1

if ˛  y  ˇ;

0

otherwise:

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D. Goldfeld

It immediately follows that for T > 0, Arthur’s truncated Eisenstein series ƒT E.z; s/ is given by X 

ƒT E.z; s/ D E.z; s/ 

 Im. z/s C .s/ Im. z/1s :

(2)

 2 1 n Im.z/T

3 Fourier Expansion of Arthur’s Truncated Eisenstein Series We now compute the Fourier expansion of ƒT E.z; s/. The proof of the Fourier expansion requires the following lemma. Lemma 3.1 Let v 2 C with 0: Then 1 2i

Z

2Ci1

2i1

(

Yw dw D w1Cv

.log Y/v .1Cv/

if 1  Y;

0

if 0  Y < 1:

Proof of Lemma 3.1 For 0 and 0 we have wv D

1 .v/

Z

1

ewx xv

0

dx : x

It follows that 1 2i

Z

2Ci1 2i1



 Z 1 1 dx dw ewx xv .v/ 0 x 2i1 Z 1 Z 2Ci1  w 1 Y dw dx 1 D xv  .v/ 0 2i 2i1 ex w x Z log Y .log Y/v 1 dx D :u t D xv .v/ 0 x .1 C v/

Yw 1 dw D w1Cv 2i

Z

2Ci1

Yw  w

It immediately follows from (2) and Lemma 3.1 that ƒT E.z; s/ D E.z; s/  ET .z; s/  .s/ET .z; 1  s/; where we have defined 1 E .z; s/ WD lim v!0 2i T

Z

2Ci1

2i1

T w E.z; s C w/ dw: w1Cv

Warning: The function ET .z; s/ is not the naive truncation of the Eisenstein series that we considered earlier in the introduction.

Arthur’s Truncated Eisenstein Series for SL.2; Z/ and the Riemann Zeta. . .

89

Theorem 3.2 (Fourier Expansion of ET ) Let T > 0; and x C iy 2 h. Let .n/ denote the Mobius function. Then Z

1

ET .x C iy; s/e2inx dx

0

D

8 ˆ ˆ ys  charŒT; 1 .y/ C ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 0 ˆ
0 and z D x C iy 2 h. Then the constant term of ƒT E.z; s/ is given by Z

1 0

    ƒT E.z C u; s/ du D ys C .s/y1s  1  charŒT; 1 .y/

 charŒ0; 1=T .y/  Z

1 2i

 T w  .s C w/y1sw C .s/.1  s C w/ysCw dw: w

2Ci1

 2i1

Proof of Theorem 3.2 With the Fourier expansion (1), we compute Z

1

ET .x C iy; s/e2inx dx

0

D

8 ˆ lim ˆ ˆ ˆv!0
1, then the integral on the right above is easily seen to be zero by shifting the line of integration to the right. This is the reason why we can multiply the integral by charŒ0; 1=T .y/. This establishes the first case of Theorem 3.2. Next, we consider the integral involving the K-Bessel function. To simplify the computation we make use of the following integral representation:  sCw 12 Z 1  .s C w/ 4jnjy cos.t/ KsCw 1 .2jnjy/ D p  sCw dt: 2 2  0 t C 4 2 n2 y2 It follows that Z

p 2 sCw y T w 1

12s2w .n/jnjsCw 2 1Cv v!0 .s C w/.2s C 2w/ 2i1 w   KsCw 1 2jnjy dw 1 2i

lim

2Ci1

2

1 D lim v!0 2i Z

1

0

Z

2Ci1

2i1

T w 22sC2w  2sC2w1 ysCw X 12s2w 2sC2w1 d jnj w1Cv .2s C 2w/ djn

cos.t/  sCw dt dw t2 C 4 2 n2 y2

2s1 s

D lim 2.2jnj/

y

v!0

1  2i

1 X .`/ X `D1

Z

2Ci1

2i1

`2s

d

12s

1 tD0

djn

4 2 n2 y   T`2 d2 t2 C 4 2 n2 y2

Z

!w

cos.t/  s 2 t C 4 2 n2 y2

dw dt: w1Cv

Now, 1 lim v!0 2i 8 1;

otherwise.

The condition 2 2 n2 y   >1 T`2 d2 t2 C 4 2 n2 y2

!w

dw w1Cv

Arthur’s Truncated Eisenstein Series for SL.2; Z/ and the Riemann Zeta. . .

91

is equivalent to yT`2 d2  1

0  t2  4 2 n2 y2

and



 1  1 : yT`2 d2

It follows that Z

p 2 sCw y T w 1

.n/jnjsCw 2 1Cv .s C w/.2s C 2w/ 12s2w w 2i1   KsCw 1 2jnjy dw

1 lim v!0 2i

2Ci1

2

D 2.2/2s1 ys

1 XX

jnj2s1

djn `D1 yT`2 d2 1

Z

q 2jnjy

1 yT`2 d2

1

tD0

1s

D 2y

1 XX djn `D1 yTd2 `2  1

d

.`/ 12s d `2s

cos.t/  s dt 2 t C 4 2 n2 y2 12s

.`/ `2s

Z

q

1 Tyd2 `2

1

0

cos.2nyt/ dt: .t2 C 1/s

If n ¤ 0 and yT > 1 it is not possible to have integers d; `  1 such that yTd2 `2  1: Therefore the above integral vanishes and the second case of Theorem 3.2 has been proved. The third case of Theorem 3.2 follows immediately from the above. t u Remarks In [2], Arthur defined truncated Eisenstein series in great generality. The explicit computation of the Fourier expansion of Arthur’s truncated Eisenstein series for higher rank groups, such as SL.n; Z/ with n > 2, seems out of reach at present. This is not the case for SL.2; Z/ because of the existence of so many special known integral representations for Bessel functions. Our knowledge of higher rank Whittaker functions is much more limited at present.

4 Maass–Selberg Relation for SL.2 ; Z/ In this section we compute the inner product of two truncated Eisenstein series which is called the Maass–Selberg relation. Arthur’s proof of the Maass–Selberg relation just uses the Rankin–Selberg method [9]. It is an algebraic proof based on the fact that if a group G acts on a topological space X then the union of all translates

92

D. Goldfeld

g  .GnX/ (with g 2 G) is just the space X: More generally for a subgroup H  G, we also have [

g  .GnX/ D HnX:

g2HnG

Arthur’s algebraic proof stands in stark contrast to the original analytic proof of Maass and Selberg which required differential operators and Green’s theorem. Theorem 4.1 (Maass–Selberg Relation) Let s; s0 2 C and T  1: Then 0 0 0 D E T sCs0 1 T ss T s s T 1ss ƒT E. ; s/; E. ; s0 / D C.s/ : C.s0 / C.s/.s0 / sCs0 1 ss0 s0 s 1ss0

Proof of Theorem 4.1 It is clear that D E ƒT E. ; s/; ƒT .E. ; s0 / “   dxdy ƒT E.z; s/  E.z; s/  ET .z; s/  .s/ET .z; 1  s/ : D y2 nh To simplify the computation of (3) we require a lemma. D E D E Lemma 4.2 We have ƒT E. ; s/; ƒT .E. ; s0 / D ƒT E. ; s/; .E. ; s0 / : Proof of Lemma 4.2 By (3), it is enough to prove that “   dxdy ƒT E.z; s/  ET .z; s0 / C .s0 /ET .z; 1  s0 / D 0: y2 nh This can be shown as follows: “   dxdy ƒT E.z; s/  ET .z; s0 / C .s0 /ET .z; 1  s0 / y2 nh “ D ƒT E.z; s/ nh

X  21 n

D

X 21 n

“

  dxdy Im. z/s0 C .s0 /Im. z/1s0  charŒT; 1 .Im. z// y2

“

T

ƒ E.z; s/  1 .nh/



ys0

C

.s0 /y1s0

  charŒT; 1 .y/

  dxdy ƒT E.z; s/ ys0 C .s0 /y1s0  charŒT; 1 .y/ 2 y 1 nh Z 1Z 1   0 dxdy 0 D ƒT E.z; s/ ys C .s0 /y1s  charŒT; 1 .y/ 2 : y 0 0 D

dxdy y2

(3)

Arthur’s Truncated Eisenstein Series for SL.2; Z/ and the Riemann Zeta. . .

93

This procedure is termed: “unravelling ET .z; s/ C .s/ET .z; 1  s/”. The x-integral above picks off the constant term of ƒT E.z; s/ which is given in Theorem 3.3 and equals    s  1 y C .s/y1s  1  charŒT; 1 .y/  charŒ0; 1=T .y/  2i Z 2Ci1 w   T .s C w/y1sw C .s/.1  s C w/ysCw dw:  w 2i1 Since   1  charŒT; 1 .y/  charŒT; 1 .y/ D charŒ0; 1=T .y/  charŒT; 1 .y/ D 0 for T  1 the proof of the lemma follows immediately. t u To complete the proof of the Maass–Selberg relation we apply Lemma 4.2 and unravel ƒT E.z; s/: We assume for the moment that 1 and 0 such that C  3 log 2 C jtj

j.1 C 2it/j > for all t 2 R:

Proof of Theorem 5.1 Let T > 1, 0 <  < 1 be fixed and t 2 R with jtj large. The main idea of the proof is to evaluate the following integral: I .t; / WD j.1 C 2it/j T

2

Z 

1

Z 0

1

ˇ ˇ T ˇƒ E.z; 1=2 C it/ˇ2 dxdy y2

in two different ways. First computation of I T .t; / (Upper Bound): We will prove the upper bound  j.1 C 2it/   log.2 C jtj/2 C 2 log.T/ : 

I T .t; /

(4)

To prove (4) note that I T .t; / D j.1 C 2it/j2

“

ˇ ˇ2 dxdy N.z; / ˇƒT E.z; 1=2 C it/ˇ y2 nh

where ˇ  ˚ N.z; / WD #  2 1 n ˇ Im. z/ >  : Since 0 <  < 1, we have the bound N.z; /

Maass–Selberg relation “

1 

which may be combined with the

0 2it 2it ˇ T ˇ ˇƒ E.z; 1=2 C it/ˇ2 dxdy D 2 log.T/   .1=2 C it/ C .1=2 C it/T C .1=2 C it/T : 2 y  2it nh

This gives the upper bound (4) . Second computation of I T .t; / (Lower Bound): Assume  D T 1 : We will prove the lower bound I T .t; / 

1 1  :  log.2 C jtj/

It follows from Theorem 3.2 that if y  T 1 then Z 0

1

ET .x C iy/e2inx dx D 0:

(5)

96

D. Goldfeld

Recall that ƒT E.z; s/ D E.z; s/  ET .z; s/  .s/ET .z; 1  s/: It follows that if  D T 1 then the nonzero Fourier coefficients of E.z; s/ coincide with the nonzero Fourier coefficients of ƒT E.z; s/ for y  : Hence I .t; /  T

1 X

j 2it .m/j

Z

1 m

mD1



2

ˇ ˇ ˇ K .2y/ ˇ2 dy ˇ ˇ it ˇ ˇ 1 ˇ  2 C it ˇ y

1 X j 2it .m/j2 jtj jtj m



4

1 X j 2it .p/j2 jtj jtj p

4

Using sieve theory one may show that j 2it .p/j  1 for a positive proportion of primes. We thus obtain the lower bound (5). There is a loss of one logarithm in the lower bound because (for large x) the number of primes  x is asymptotic to x= log x by the prime number theorem. Theorem 5.1 immediately follows by comparing the lower bound (5) with the upper bound (4). t u

References 1. J. Arthur, A trace formula for reductive groups I. Terms associated to classes in G(Q). Duke Math. J. 45, 911–952 (1978) 2. J. Arthur, A trace formula for reductive groups. II: applications of a truncation operator. Compos. Math. 40(1), 87–121 (1980) 3. F. Brumley, Effective multiplicity one on GLN and narrow zero-free regions for RankinSelberg L-functions. Am. J. Math. 128(6), 1455–1474 (2006) 4. B. Casselman, Truncation Exercises, Functional Analysis VIII. Various Publications Series (Aarhus), vol. 47 (Aarhus University, Aarhus, 2004), pp. 84–104 5. P. Garrett, Simplest Example of Truncation and Maass-Selberg Relations, Vignettes (2005), http:/www.math.umn.edu/~garrett/ 6. P. Garrett, Truncation and Maass-Selberg Relations, Vignettes (2005), http:/www.math.umn. edu/~garrett/ 7. S.S. Gelbart, E.M. Lapid, Lower bounds for L-functions at the edge of the critical strip. Am. J. Math. 128(3), 619–638 (2006) 8. S.S. Gelbart, E.M. Lapid, P. Sarnak, A new method for lower bounds of L-functions. C. R. Math. Acad. Sci. Paris 339(2), 91–94 (2004) 9. D. Goldfeld, Automorphic Forms and L-Functions for the Group GL.n; R/ (Cambridge University Press, Cambridge, 2006) ˝ 10. D. Goldfeld, X. Li; A Standard Zero Free Region for RankinUSelberg LFunctions. Int. Math. Res. Notices (2017). doi:10.1093/imrn/rnx087

Arthur’s Truncated Eisenstein Series for SL.2; Z/ and the Riemann Zeta. . .

97

11. D. Hejhal, On a result of G. Polya concerning the Riemann -function. J. d’Analyse Math. 55, 59–95 (1990) 12. H. Jacquet, J.A. Shalika, A non-vanishing theorem for zeta functions of GL n . Invent. Math. 38(1), 1–16 (1976/1977) 13. J.C. Lagarias, M. Suzuki, The Riemann hypothesis for certain integrals of Eisenstein series. J. Number Theory 118, 98–122 (2006) 14. R.P. Langlands, in Algebraic Groups and Discontinuous Subgroups. Proceedings of Symposia in Pure Mathematics, Boulder, CO, 1965 (American Mathematical Society, Providence, RI, 1966), pp. 235–252 15. R.P. Langlands, The trace formula and its applications: an introduction to the work of James Arthur. Can. Math. Bull. 44(2), 160–209 (2001) 16. H. Maass, Uber eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann. 121, 141–183 (1949) 17. P. Sarnak, Nonvanishing of L-functions on 2 after (24), we obtain that the contribution of n 6 N1 to I1 in (25) will be O" .T 3=4C" /. Since Jn .T/ T 1=2 [see (31) and (32)] it transpires that the condition N1 6 n 6 N0 may be replaced by T1 6 n 6 T0 , where T0 ; T1 are defined by (12). In this process the total error term will be O" .T 3=4C" /. Therefore, for the range 0 < U 6 T 1=2" , we have uniformly I1 D eiU=2 e3i=8

X

h.n; U/n1=2 Jn .T/ C O" .T 3=4C" /;

(29)

T1 6n6T0

and the range of summation in (29) is easier to deal with than the range of summation N1 < n 6 N0 . At this point we shall evaluate Jn .T/ by one of the theorems related to exponential integrals with a saddle point, e.g., by the one on p. 71 of the monograph by Karatsuba–Voronin [13]. This says that, if f .x/ 2 C.4/ Œa; b, Z

b a

e2if .c/ e2if .x/ dx Dei=4 p C O.AV 1 / f 00 .c/    p  1 p 1 C O min.jf 0 .a/j ; a / C O min.jf 0 .b/j ; b / ;

where 0 < b  a 6 V; f 0 .c/ D 0; a 6 c 6 b; f 00 .x/ A1 ; f .3/ .x/ .AV/1 ; f .4/ .x/ A1 V 2 : We shall apply (30) with f .x/ D

1 fn .x/ ; c D tn ; a D 12 T; b D T; V D 12 T; f 00 .x/ ; 2 T

(30)

108

A. Ivi´c

so that A D T; f .3/ .x/ T 2 ; f .4/ .x/ T 3 , which is needed. The term O.AV 1 / in (30) will make a contribution which is O" .T 3=4C" /, and the same assertion will hold for the other two error terms in (30). This will lead to X

I1 D eiU=2 e3i=8

eifn .tn / h.n; U/n1=2 ei=4 q C O" .T 3=4C" /: fn00 .tn / 2

T1 6n6T0

(31)

Since, by (24) and (27), n o tn .tn C U/ 3 C 12 .tn C U/ log  2 tn  tn log n eifn .t/ D exp i tn log 2 2 n o 2 t .tn C U/ .tn C U/ 3 1  D exp 12 itn log n C Ui log it  it log n n n 2 2 8 3 2 o n .tn C U/ 3  2 itn D exp 12 iU log 2   tn C U iU=2 ; D e3itn =2 2 it follows that p 2eiU=2 ei=8 s   X 2tn .tn C U/ 3itn =2 tn C U iU=2 1=2 e  h.n; U/n : 3tn C 2U 2 T 6n6T

I1 DO" .T 3=4C" / C

1

(32)

0

In view of (27), (28) and n T 3=2 we have s

s 2t .t C U/ 2.2/3 n n D n1=2 n1=2 3tn C 2U tn .3tn C 2U/   p D 2.2/3 n1=2 tn1 31=2 C O.U=T/ r  1 2 .2/3 n1=2 .8 3 n2 /1=3 C O.U/ D C O.UT 5=4 / 3 r  1=6 n C O.UT 5=4 /: D2 3 It remains to evaluate iU=2 3itn =2

A WD A .n; U/ D e

e



tn C U 2

iU=2 :

(33)

On a Cubic Moment of Hardy’s Function with a Shift

109

To achieve this we need a more precise expression for tn than (28). Putting (28) in (27) we first have tn3 D 8 3 n2  Utn2 D 8 3 n2  4 2 Un4=3 C O.n2=3 U 2 /;  1=3  1 C O.U 2 n4=3 / tn D .8 3 n2  4 2 Un4=3 / D 2n2=3  13 U C O.U 2 n2=3 /: This yields the second approximation tn D 2n2=3  13 U C O.U 2 n2=3 /:

(34)

Inserting (34) in (27) we shall obtain next tn D 2n2=3  13 U 

U 2 n2=3 18

C O.U 3 n4=3 /:

(35)

In fact, (35) is just the beginning of an asymptotic expansion for tn in which each term is by a factor of Un2=3 of a lower order of magnitude than the preceding one. This can be established by mathematical induction. Then we see that (c’s and d’s are effectively computable constants) n 3 U 2 n2=3 log A D i  12 U  .2/n2=3 C 12 U C C c3 U 3 n4=3 C O.U 4 n6=3 / 2 12  o U 2 n2=3 U 3 4=3  C 12 U log n2=3 C C O.U n / 3 36 2 D 3in2=3 C 13 iU log n C ic2 U 2 n2=3 C ic3 U 3 n4=3 C O.U 4 n6=3 /; and therefore 2=3

A D niU=3 e3in

˚  1 C id2 U 2 n2=3 C id3 U 3 n4=3 C O.U 4 n6=3 / ;

(36)

and we remark that the last term in curly brackets admits an asymptotic expansion. Since U T 1=2" by assumption, U 2 n2=3 T 2" and thus each term in the expansion will be by a factor of at least T 2" smaller than the preceding one. One sees now why the interval of integration in (10) and (11) is ŒT=2; T and not the more natural Œ0; T, because the terms dj U j n2j=3 become large if n is small, and K.n; U/ is then also large. Inserting (36) (as a full asymptotic expansion) and (33) in (32), we obtain the assertion of Theorem 1, having in mind that we have shown that jI2 j C jI3 j " T 3=4C" :

110

A. Ivi´c

To prove Theorem 2 note that, with S.˛; N/ defined by (14), we have 0 1 ZB X Z B X 2=3 2=3 jS.˛; N/j2 d˛ D @ d32 .n/ C d3 .m/d3 .n/ei˛.m n / A d˛ A

n N

A

m¤n N

X

D .B  A/

0

X

d32 .n/ C O @

n N

m¤n N

1

d3 .m/d3 .n/ A : jm2=3  n2=3 j

(37)

Note that the function d3 .n/ is multiplicative and d3 .p/ D 3. Thus for 1 and p a generic prime we have 1 X nD1

d32 .n/ns D

Y

1 C d32 .p/ps C d32 .p2 /p2s C   



p

D  9 .s/

Y

.1  ps /9

p

 Y 1 C 9ps C d32 .p2 /p2s C    p

D  9 .s/F.s/;

(38)

where F.s/ is a Dirichlet series that is absolutely convergent for 1=2. Thus the series in (38) is dominated by  9 .s/ and it follows, by a simple convolution argument, that X d32 .n/ D C1 x log8 x C O.x log7 x/ (39) n6x

for a constant C1 > 0. Next, by using the trivial inequality jabj 6

1 2 .a 2

C b2 /

and (39), it is seen that the double sum in (37) in the O-term is, by symmetry,

N 1=3

X d2 .m/ C d2 .n/ 3 3 jm  nj

m¤n N

D 2N 1=3

X m N

d32 .m/

X n¤m;n N

1 jn  mj

N 4=3 log9 N; since, for a fixed m such that m N, X n¤m;n N

1

log N: jn  mj

On a Cubic Moment of Hardy’s Function with a Shift

111

From this estimate and (39) it follows that Z

B

jS.˛; N/j2 d˛ N 4=3 log9 N;

(40)

A

which implies the assertion of Theorem 2. A further discussion on the cubic moment of Z.t/ is to be found in Chap. 11 of the author’s monograph [8]. Remark 6 An interesting problem is to determine the true order of the integral in (40). Remark 7 If the divisor function d3 .n/ is not present in the definition (14), namely if one considers the sum X 2=3 e˛in .˛ ¤ 0/; T.˛; N/ WD N 0 is arbitrarily small but fixed, we have  

0 1 jsj ; .s/ D  log CO

2 jsj

.jtj  1; sayI 1   2 /;

(16)

where 1 < 2 are any fixed real numbers. Under the same conditions one also has 

d ds

n

0 .s/ jsjn ;

.n  1/:

Some further facts concerning .s/ and .s/ will be given in Sects. 5 and 6.

(17)

Analogues of Pair Correlation of Zeta Zeros

117

Notation Throughout this paper, T will denote a large number tending to 1,  will be a fixed positive number which can be taken arbitrarily small and which may not have the same value each time it appears, and letters with indices such as Cj will denote sufficiently large positive constants of fixed value. We write N D ZC [ f0g. We will employ the Iverson notation that for a statement S, ŒS D 1 if S is true, and ŒS D 0 if S is false.

2 A Sketch of Montgomery’s Derivation The explicit formula from Landau’s book [16] is that, if x > 1 and x ¤ pk (p prime, and k a positive integer), then X .n/ nx

ns

1

X xs X x2rs x1s 0  C D  .s/ C  1s   s rD1 2r C s 

(18)

provided s ¤ 1; s ¤ ; s ¤ 2r. Here  runs through the non-trivial zeros of .s/, X xs . and the series over  is convergent subject to the interpretation lim U!1 s j= jU

Upon assuming RH, Montgomery deduced that .2  1/

X

xi



.  12 /2 C .t   /2

D x

 1 Cit X   Cit ! x x .n/ .n/ n n n>x nx

X

 12

1



1 .2  1/x 2 0 .1  C it/x 2  Cit C  .  1 C it/.  it/ 1

x 2

1 X rD1

.2  1/x2r ; .  1  it  2r/. C it C 2r/

(19)

valid for > 1, and all x  1. In this formula Montgomery used (13)–(16) to 0 0 replace  .1  C it/ by   .  it/  log.jtj C 2/ C O.1/ D  log.jtj C 2/ C O .1/ (for s in a fixed strip in > 1), and he put upper-bound estimates for the last two terms of (19). Montgomery took D 32 , squared the modulus of both sides, and then integrated both sides over t from 0 to T. From the expression thus arising from the 3 left-hand side of (19), he discarded those  62 Œ0; T within an error of O.log R 1 T/, and in order to evaluate the integral he extended the range of integration to 1 within an even smaller error. In this way the left-hand side of (19) led to the expression (2). To carry out the integration of the square of the series involving .n/ coming from the right-hand side of (19), Montgomery resorted to Parseval’s formula for Dirichlet series from [18]: If

1 X nD1

njan j2 converges; then

Z

T 0

ˇX ˇ2 X ˇ ˇ an nit ˇ dt D jan j2 .T C O.n//; ˇ n

The result of this calculation was (4) (or (7)).

n

118

Y. Karabulut and C.Y. Yıldırım

3 A Modified Approach With D 52 , (19) reads X 

  32 Cit X   52 Cit ! x x .n/ C .n/ n n nx n>x

X

4xi 1 D x 2 2 4 C .t   / 

0 

  1

4x 2

 1 3 4x 2  5  C it x2Cit C  3 2 C it 2  it 2

1 X rD1

3 2

x2r  :  2r  it 52 C 2r C it

Using 0 1 0 .1  C it/ D  .  it/ C log  C   .  it/.1  C it/ ! !! 1  0 3  C it 0

 it  C 1C 2  2  2

(20)

which follows from the formulas in Chap. 12 of [2], with D 52 , we have X 

X X 4xi.t/ 3 5 2 2 it  x2 D x .n/n .n/n 2 it 4 C .  t/2 nx n>x !   0 5 2   it  log  Cx  2    ! x2  0 1  0 9 it it C C C  2  4 2  4 2 1

x2 4x 2 it  C   C 5 3  it 32  it C it 52  it 2 2 1

4x 2 it

1 X rD1



3 2

x2r  :  2r  it 52 C 2r C it

(21)

Analogues of Pair Correlation of Zeta Zeros

119

Using (15), and doing elementary estimations, we can simplify (21) into X 

X X 4xi.t/ 3 5 2 2 it  x2 D x .n/n .n/n 2 it 4 C .  t/2 nx n>x   Cx2 log.jtj C 3/ C O.1/ ! ! 1 5 x 2 x2 CO CO ; .x  1/: .jtj C 1/2 jtj C 1

(22)

When t is let to run through a set we sum both sides of (22) over this set of values X of t. This will be feasible if one can calculate the sums over t, in particular piat t

where p is a prime and a is a natural number.

4 Pair Correlation of Zeta Zeros We apply our method first to the quantity (2) considered by Montgomery. So, letting t run through those ordinates Q of the zeros of the Riemann zeta-function which are in the interval .0; T, from (22) we have X X 

0x 0 0; 1 C n C 2 4 4

.t > 0/;

(50)

i.e. # 0 .t/ is strictly increasing for t > 0. We find from (14) that

0 2# .0/ D 

0

    1  D  e C log 4 C C log 2  5:37 : : : ; 2 2

(51)

and we know from Sect. 6.5 of [4] that # 0 .t/ > 0;

.t  10/:

(52)

We deduce that

0 .s/ D 0; s 62 R H) s D

1 ˙ i; for a unique  2 .0; 10/ 2

( must be near 2, since from (47) we see 2# 0 .t/ D log. 2t / C O. t12 /).

(53)

124

Y. Karabulut and C.Y. Yıldırım

Hall’s paper [14] contains thorough information about the zeros and poles of Z1 .s/ (Hall denotes our Z1 .s/ by F.s/). We see that fsI s is a pole of Z1 .s/g D f0; 1; 3; 5; 7; : : :g

(54)

(the pole at s D 1 is a double pole coming from the simple poles of .s/ and of

.s/ at s D 1, the simple pole at s D 0 arises from .0/ D 0, and the simple poles at s D 3; 5; 7; : : : come from the poles of .s/ at these points). Hall, assuming RH, showed that all non-real (which will be termed as non-trivial) zeros of Z1 .s/ are on the critical line, so that, assuming RH, the non-real points where Z1 .s/ vanishes are the relative maxima of j. 21 C it/j, and the multiple zeta zeros if ever these exist. Some information about these have already been given in the beginning of this section. Hall also proved assuming RH that the number of the zeros of Z1 .s/ (counted according to multiplicity) with ordinates in .0; T, NZ1 .T/, is Z0 3 1 NZ1 .T/ D N .T/  sgn .t/ C ; 2 Z 2

(55)

provided that T is not the ordinate of a zero of .s/ or of Z1 .s/. There are real zeros of Z1 .s/, all of them simple zeros. These may be termed as the trivial zeros of Z1 .s/. The trivial zeros are placed symmetrically with respect to the point 12 because of the functional equation Z1 .s/ D  .s/Z1 .1  s/;

(56)

which follows from (32) and (11). We have fsI s is a trivial zero of Z1 .s/g D fz0 ; z` ; 1  z` I ` 2 ZC g; z0 D

1 ; z1 2 .3; 5/; z2 2 .5; 7/; z` 2 .2` C 2; 2` C 3/ 2 for `  3: (57)

Since Z1 .s/ doesn’t have an associated Dirichlet series, we will need a Dirichlet Z0 polynomial which approximates 1 .s/ in > 1; jtj  t0 . From (32) we have Z1

Z10 .s/ D Z1

0 .s/ 



 00 .s/

0 

.s/ 2

1

C

0 .s/ .s/ 

2

0



0

0

.s/

0

.s/

:

(58)

Here we use the estimates (16), (17), and  .j/ .s/ .log log jtj/j ; 

.j D 1; 2I  1; jtj  t0 /;

(59)

Analogues of Pair Correlation of Zeta Zeros

125

which depends on RH and follows from Corollary 13.14 of [19] by a standard application of Cauchy’s estimate, so that we can write Z10 .s/ D Z1

0 .s/ 

C

1C

2  00 jtj  .s/ log 2

2 0 jtj  .s/ log 2

 CO

 1 : jtj log jtj

(60)

Expanding the denominator as geometric series, we see that log jtj

b log log jtj c

1 1C

2 0 jtj  .s/ log 2

D

X



kD0

 CO jtj

1

2 jtj log 2

0 .s/ 

!k



C1 log jtj log log log jtj exp log log jtj

(61)  :

Now, for k 2 N and > 1, we write  0 k 1 X  .k/ .n/  .s/ DW ;  ns nD1

(62)

 0 k 00 1 X   .k/ .n/  .s/ : .s/ DW   ns nD1

(63)

Using (61)–(63) in (60) gives the following Dirichlet polynomial approximation (the series begins with m D 2 since .kC1/ .1/ D .k/ .1/ D 0). Proposition 1 Assume RH. In the region 1 <  0 ; jtj  t0 , we have log jtj ! !k b log log jtj c 1 X Z10 1 X 2 2 .k/ .kC1/ .s/ D .m/   .m/ jtj jtj Z1 ms kD0 log 2 log 2 mD2    C1 log jtj log log log jtj 1 CO jtj exp : log log jtj

(64) 0

Remark We were led to the approximation (64) because the idea of replacing 

.s/ T by log 2 in (32) and then developing an approximating Dirichlet series, as was done 2 in [1], ends up giving a weaker (valid only for x up to T 3  ) result than Theorem 1 0 2 1 in Sect. 7. Unconditionally, we have  .s/ j .log t/ 3 .log log jtj/ 3 ; .  1; jtj  t0 / at our disposal (see [21], Sect. 6.19), causing the error term in (64) to become 1 O.jtj 3 C /, and this in turn also leads to a smaller range in Theorem 1. Anyhow, results of the type we are pursuing make more sense when RH is assumed.

126

Y. Karabulut and C.Y. Yıldırım

6 Preliminaries Concerning the Iterated Convolutions of the von Mangoldt Function We shall need in Sects. 7 and 8 some properties of the arithmetic functions .k/ .n/ and .k/ .n/. We note that the article [5] by Farmer et al. contains some related results. First, proceeding inductively and denoting Dirichlet convolution by , we see the trivial bound, for k  1, n X X  .log n/k1 .k/ .n/ D .k1/ .n/ D .d/.k1/ .d/ D .log n/k : d djn

djn

(65)

Next, using (65), we see that .k/ .n/  .log n/kC2 ;

.k D 0; 1; 2; : : :/;

(66)

by dint of .k/ D .k/ 2 ; where

1 X X 2 .n/  00 .s/ DW .< s > 1/; and 2 .d/ D .log n/2 : s  n nD1

djn

(67)

Note that by comparing coefficients in

 00 .s/ 

D



0 .s/ 

0

C



0 .s/ 

.0/ .n/ D 2 .n/ D .n/ log n C .2/ .n/ D .n/ log n C

2

X djn

we have

n : .d/ d (68)

It is easy to calculate that for a 2 ZC and p a prime number, 2 .pa / D .2a  1/.log p/2 ! a  1 .k/ .pa / D .log p/k ; k1 " ! !# a a1 .k/ a .p / D 2  .log p/kC2 : kC1 k Here we are employing a convention that for u; v integers which are  1, 8 ˆ 1 if r D s; ˆ ˆ ˆ ! ˆ < 0 if s D 1 and r  0; r WD 0 if r < s; ˆ s ˆ ˆ ˆ rŠ ˆ : otherwise: sŠ.r  s/Š Now we give two lemmas which will be used in Sect. 7.

(69) (70)

(71)

(72)

Analogues of Pair Correlation of Zeta Zeros

127

Lemma 1 Let k 2 ZC . We have, for N  1, X .k/ .N  n/ .log log 3N C 7/k

.log 2N/k : n .k  1/Š N

(73)

n 2

The same bound holds with .k/ .N C n/ in place of .k/ .N  n/. Here .K/k WD

k1 Y

.K C j/;

(74)

jD0

with the convention .K/0 WD 1. The k D 1 case of this lemma was shown by Gonek [13] within his proof of (25). Proof Let ˇ ˇ ˇ˚ ˇ S.x/ WD ˇˇ N  x < n  N W .k/ .n/ ¤ 0 ˇˇ: By (65) we can write Z N X .k/ .N  n/ 2 dS.x/ k  .log 2N/ D .log 2N/k  n x 1 N

n 2

(

S.x/ ˇˇ N2 ˇ C x 1

Z

N 2

1

S.x/ dx x2

)

We now need an upper bound for S.x/. To this end we quote the following result of Tudesq [22]: X

1

y 1; .d0 ; p/ D 1/

(81)

130

Y. Karabulut and C.Y. Yıldırım

which follows from (68), and the case  D 0, we obtain .k; 1I pn/  k.log p/.log pn/k1

X

2 .d/

djn

C2.log p/.log n0 /k

X

.d0 / C .2r C 1/.log p/2 .log n0 /k :

d0 jn0

Now the assertion in the case  D 1 and t u X.p; n/ > 1 is immediately seen. .k; I bn/, where b is either 1 or a prime We now begin examining the sum n bx log x , log 2

then

Lemma 4 Let b be 1 or any prime number p  x. For 1 C Œb ¤ 1  k C   there exists a constant C2 > 1 such that

log x , log 2

number p  x (clearly, the sum is void if b > x), for k  .k; I bn/ D 0). The results will form

X

.k; I bn/ 

n bx

log x log 2

C2k x .log x/kC21 : .k C 2  1/Š b

(if k >

(82)

Proof First we take b D 1. We have X

.k; I n/ 

nx

2 x.log x/kC21 ; .x ! 1; any fixed k 2 N;  2 f0; 1g; k C   1/ .k C 2  1/Š (83)

as a generalization of the prime number theorem, since " RessD1

0  .s/ 

k

 00 .s/ 



# xs 2 x.log x/kC21 ; .k 2 N;  2 f0; 1g; k C   1/  s .k C 2  1/Š (84)

(the details of the proof of (83) are similar to the zeta-function based proof of the prime number theorem as in [2]). However, we will also need to work with k not fixed. We want to show that X nx

.k; I n/ 

C3k x.log x/kC21 .k C 2  1/Š

(85)

with some absolute constant C3 > 1 by doing induction on k. The induction basis cases are covered in (83). Starting from the definition in (77) we see that

Analogues of Pair Correlation of Zeta Zeros

X

.k C 1; I n/ D

nx

XX nx djn

131

 X X n .d/ D  k; I .d/ .k; I e/ d x dx e d



X .d/  C3k x x kC21 log .k C 2  1/Š dx d d



C3kC1 x.log x/kC2 ; .k C 2/Š

(86)

where we have used X .n/    x j .log x/jC1 C O .log x/j ; log D n n jC1 nx

.j 2 N/

(87)

which can be seen from Mertens’s classical result via partial summation (this result is included in Lemma 5.5 of [5]). This proves (85). X Next, taking b D p a prime number, we examine the sum .k; I pn/. Let ` n px

be the unique integer such that x 2 Œp` ; p`C1 /. First we examineX the cases k C  D 1 which turn out to be special. For .k; / D .1; 0/, the sum is .pn/. The nonn px

zero summands are from n D 1; p; p2 ; : : : ; p`1 , and the sum has value ` log p 2 . 12 log x; log x, so that X

.1; 0I pn/ D

n px

For .k; / D .0; 1/, the sum is

X

.pn/  log x:

(88)

n px

X

2 .pn/. The non-zero summands are from n D

n px

1; p; p2 ; : : : ; p`1 with total contribution `2 .log p/2 2 . 14 .log x/2 ; .log x/2 , and from x n D pr pr11  with a prime p1 ¤ p and r  0; r1  1. By p 2 .pa11 pa22 / D 2 log p1 log p2 ;

.a1 ; a2  1/;

(89)

which is seen from (68), and .x/ WD

X px pW prime

log p  C4 x;

.x > 0/

(90)

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Y. Karabulut and C.Y. Yıldırım

1 (with C4 D 1 C 36260 according to Dusart [3]), we have XX X X 2 .prC1 pr11 /  2 log p log p1 r   r1 r0 r1 1 pr p11  px x prC1

p1 

r0;r1 1

1

X X  x  r1  2C4 log p prC1 r0 r 1 1

1

X x x  C5 log p  C6 log p; rC1 p p r0 where we have made use of log



x prC1

 r1

1

(91)

 2 for non-void sums so that r1 can be at most

x prC1

. We have chosen to use the majorization (90) because if we try to evaluate log 2 all of the sums asymptotically we will get quantities containing the factor log px and this will necessitate examining the results according to the size of p relative to x and thus further complicate the matter. So, we express the result for this case as X

.0; 1I pn/ D

n px

X n px

x 2 .pn/  C7 log p C .log x/2 : p

(92)

We wish to obtain an inequality of the type (85) by induction on k C , but from (88) and (92) we see that in the k C  D 1 cases the power of log x won’t match the power in (85) if logx x D o.p/; p  x. So, for the induction basis we take k C  D 2 which has the two cases .k; / D .2; 0/X and .k; / D .1; 1/. X For .k; / D .2; 0/, the sum is .2/ .pn/, differing from 2 .pn/ only in the n px

n px

contribution from pn D p; p2 ; : : : ; p` . This contribution, p p > x. Hence, from (92) we have X

.2; 0I pn/ D

n px

X n px

`.`1/ .log p/2 , 2

vanishes for

x .2/ .pn/  C8 log x: p

Now take .k; / D .1; 1/ in which case the sum is

X

(93)

.1/ .pn/. The non-zero

n px

summands are from n D p; p ; : : : ; p with total contribution `.`1/.2`1/ .log p/3 6 r r1 r r1 r2 by (71), and from n D p p1 ; .r  0; r1  1/ and n D p p1 p2 ; .r  0; r1 ; r2  1/ where p; p1 and p2 are distinct primes. We have, by (67), (69) and (89), 2

`1

.1/ .prC1 pr11 / D .4r C 1/.log p/2 .log p1 / C .4r1  3/.log p1 /2 .log p/ .1/ .prC1 pr11 pr22 / D 6.log p/.log p1 /.log p2 /:

(94) (95)

Analogues of Pair Correlation of Zeta Zeros

133

Then, estimating similarly to (91), we have X

.1/ .prC1 pr11 /  .log p/2

r pr p11  px r0;r1 1

X X .4r C 1/ r0

X 

r1 p1 

C.log p/

XX

x prC1

 p1 

X

1

XX

 .4r1  3/

 C9 .log p/2

.4r C 1/

r0

 r1

x



1

log

prC1

r0 r1 1

X

x prC1

.log p1 /2 1 r1

X  x  r1 prC1 r1

.4r C 1/

r0

CC4 .log p/

1 r1

X

.4r1  3/

r0 r1 1

 C4 .log p/2

log p1 

 r1

x

1

prC1

x prC1

X X 4r1  3  x  r1 CC10 .log p/.log x/ r1 prC1 r0 r 1 1

1

x  C11 .log p/.log x/; p and X

.1/ .prC1 pr11 pr22 /  3.log p/

r r pr p11 p22  px r0; r1 ;r2 1

X  r0 r1 1 p  1

 C12 .log p/

X x 2prC1

X

 C13 x.log p/





X 1 X prC1 r 1 r0

x  C15 .log p/.log x/: p



X p1 

x 2prC1

x 2prC1

prC1 pr11 log p1 pr11

 r1

.log p1 /

x r prC1 p11

log p2 

x

X p1 

X 1 prC1 r0

X r2 1

1 r1

1

 C14 x.log p/

 p2 

log p1

x 2prC1

X

r2 1

1 r1

X

 r0 r1 1 p  1

X

log p1

1

X 1 pr1 r 1 1 1

1 r2

 r1

2

134

Y. Karabulut and C.Y. Yıldırım

Putting together these calculations, we have for the case .k; / D .1; 1/, X X x .1; 1I pn/ D .1/ .pn/  C16 .log x/2 : p x x n p

(96)

n p

Formulas (93) and (96) are of the same type as (85), and this furnishes the induction basis for establishing X

.k; I pn/ 

n px

k C17 x .log x/kC21 : .k C 2  1/Š p

(97)

with some absolute constant C17 > 1. Letting pn D de, we have X

 .k C 1; I pn/ D

n px

XX n px djpn

D

X

 pn  .d/  k; I d

.d/

d px .d;p/D1

D

X

X

.k; I e/ C

X

e dx pje

.d/

d px .d;p/D1

X

.d/

x e0  pd

.k; I e/

e dx

dx pjd

.k; I pe0 / C

X

X d0  px

.pd0 /

X

.k; I e/:

e pdx 0

(98) The last two double sums can be majorized by using the induction hypothesis and (87). We have X

X

.d/

d px .d;p/D1

.k; I pe0 / 

x e0  pd

k C17 x X .d/  x kC21 log .k C 2  1/Š p x d d d p

k C17 x  .k C 2  1/Š p



   .log x/jC2 kC21 C O .log x/ ; .k C 2/

and, since .pd0 /  .d0 /, X d0  px

.pd0 /

X

.k; I e/ 

e pdx 0

  k C17 x X .pd0 / x kC21 log .k C 2  1/Š p 0 x d0 pd0 d p



k C17 x .k C 2  1/Š p



   .log x/kC2 C O .log x/kC21 : .k C 2/

Using these in (98), the induction step is completed. Next, we prove

t u

Analogues of Pair Correlation of Zeta Zeros

Proposition 2 For 0  k1 ; k2  Sk1 ;k2 ;1 ;2 .x/ WD

X

135

log x ,  ; log 2 1 2

D 0 or 1, k1 C 1 ; k2 C 2  1, we have

.k1 ; 1 I n/.k2 ; 2 I n/

nx

D

P.k1 ; k2 ; 1 ; 2 / x.log x/k1 Ck2 C21 C22 1 .k1 C k2 C 21 C 22  1/Š ! k1 Ck2 C18 x.log x/k1 Ck2 C21 C22 2 CO ; .maxfk1 ; k2 g/Š

(99)

where 8 ˆ k1 Š ˆ ˆ ˆ ˆ ˆ ˆ ˆ2  .maxfk1 ; k2 g/Š ˆ ˆ b logloglogT T c C 1 differs from the previous case within an error a1 X

T log p

pa

log T

`Db log log T c

C

a1 `

T log p

pa

!

2

!` !`C1 ! ! a  1 a 2 log p 2 log p 4 C2 T T ` `C1 log 2 log 2

2 log p T log 2 a1 X log T

`Db log log T c

!`C1 3 5 a `C1

!

C33 log p log T

` :

Analogues of Pair Correlation of Zeta Zeros

157

!   a ae `C1 Employing the inequality , which is a simple consequence  `C1 `C1 of Stirling’s formula, the last quantity can be simplified to 

a1 X

T log pa pa

log T

`Db log log T c

T log pa

pa



C34 a log p ` log T

`

C36 log log T log T



T log pa pa



a1 X log T

`Db log log T c

 logloglogT T

a

p

C35 `

`



C37 log T log log log T exp log log T

 ;

and this is contained in an error term of the type as in (150) (if necessary by replacing C32 by a larger constant). t u We now resume the calculation for the correlation of zeta zeros with the zeta maxima on the critical line starting with using (150) in (127). From the sum over n  x we have x

2

X

2

.n/n

nx

X

%

n

T 2 x, and then using (136) to bound the resulting Dirichlet series, so that we have I1 C I2 D x

 12

X mx

L.m; 1  s/

 x 1s m

C

X

L.m; s/

m>x

!  1 x 2 C1 C51 .1 / log T CO exp : 1 C jtj log log T

 x s

!

m (178)

166

Y. Karabulut and C.Y. Yıldırım

The last step for completing the derivation of the explicit formula for Z1 .s/ is 1 Z0 replacing the term x 2 s Z11 .1  s/ in (168) with 1

x 2 s

Z10

0 jtj 1 1 1 .s/  x 2 s .s/ D x 2 s log C O .x 2  /; Z1

2

(179)

by using (139). Combining (168), (174), (177)–(179) we obtain Proposition 5 (Explicit Formula; Assuming RH) For s D C it with 32 

, jtj 2 Œ2A; T with A a sufficiently large constant, and for any arbitrarily small  11 4 fixed  > 0, we have, as T ! 1, X %

!  x 1s X  x s L.m; 1  s/ C L.m; s/ m m mx m>x !  1 x 2 C C52 ./ log T jtj 1 s CO exp C x 2 log 2 jtj log log T

.2  1/xi 1 D x 2 1 2 2 .  2 / C .v  t/

X

1

CO .x 2  /;

(180)

where L.m; s/ was defined in (177). In (180) we take D 52 and sum both sides over t D Q 2 . T2 ; T, where runs through the non-trivial zeros of Z1 .s/. In view of (55) and (1) we have X X 

T Q 2 x T

T Q 2 T M for some fixed M > 43 . Upon using (16) and trivially estimating the sum over Q by (55), we have x

2

X

X

m>T M

T Q 2 T M

X .k/ .m/ 5

m>T M

# :

m2

By (65) and (66) we see X .kC1/ .m/ m

m>T M

5 2



.log T M /kC1 T

3 2M

;

X .k/ .m/ m>T M

m

5 2



.log T M /kC2 3

T 2M

;

so that X

X

m>T M

T Q 2 T



.log T M /k1 Ck2 C1 C2 C1 : T 2M (187)

When plugged in (186) this brings an error log T

b log log T c

T

12M

log T

X

k1 ;k2 D0

2M log T T log 4 3 log T

!k1 Ck2 C1 C2

T

log T

b log log T c 12M

log T

X

.3M/k1 Ck2

k1 ;k2 D0 3

T 12M .log T/.3M/ log log T T 1 2 M ;

(188)

which is very small for M > 43 . In applying (122)–(123), the error terms of these formulas give rise to error terms of the form log T

b log log T c

T

X

k1 ;k2 D0

1 max.k1 ; k2 /Š

2C18 log x T log 4

!k1 Ck2 C` ; .` D 0; 1; 2/;

and recalling that x  T M , we see that these error terms can be bounded as

Analogues of Pair Correlation of Zeta Zeros

169

log T

b log log T c

X

T

k1 ;k2 D0

log T

b

c

log log T k1 Ck2 X Ck2 X k C57 57 C571

T max.k1 ; k2 /Š k Š 2 k k k D0 1

2

(189)

2

log T

b log log T c

X

T

k2 D0

k2 C58  eC58 T D C59 T: k2 Š

Now, going back to (184) we have X X 

T Q 2 1:

(202)

Some computing reveals that (202) is satisfied when we take D 0:78842, and we can state

Analogues of Pair Correlation of Zeta Zeros

173

Corollary 2 Assume RH. There exist infinitely many gaps of size < 0:78842 of the  ) in the sequence composed of the zeros and maxima points average gap ( log T on the critical line of .s/. Remark Since the critical zeros of  and Z1 are interlaced these small gaps are between a zero of  and a zero of Z1 . The result of Corollary 2 doesn’t seem not very strong, but it is better than the trivial bound 1, and it is the first result of its kind. Applying (198) for the proportion of simple zeros of   Z1 doesn’t give a bright result. By some elementary considerations and Hall’s result given in (55) we may take that a non-simple zero 12 C it1 of   Z1 has multiplicity at least 3. Then, Montgomery’s argument for the proportion of simple zeros, together with X 3  mt X 1 1 and a numerical fact from the proof of Corollary 1.3 of 2 0x   C x2 log q jtj C O.1/

(206)

for jtj  1 and x  1. Now let be a primitive character to the modulus q , and let t run through the ordinates  2 .1; T of the zeros of L.s; /. Summing (206) over this set of t values, we have X X  1x

X

(207)

n

11

This completes our calculation in this section, and we state our findings as Theorem 3 Assume GRH. Let 12 C i and 12 C i run through the critical zeros of L.s; / and L.s; /, respectively. Assume that the moduli q ; q are fixed. Then, as T ! 1, X

F ; .x; T/ WD

X

0 0 jS2 .q/ j

N for any " > 0, where the L1 norm is the norm on D. The main difference with previous results involving cusp forms (the first one being due to Laurinˇcikas and Matsumoto [13]) is that we do not fix one such Lfunction L.f ; s/ and consider shifts (or twists) L.f ; s C it/ or L.f  ; s/, but rather we average over the discrete family of primitive forms in S2 .q/ . It is also important to remark that the condition (1) is necessary for a function on D to be approximated by L-functions L.f ; s/ with f 2 S2 .q/ . (We will give more general statements where the discs D are replaced with more general compact sets in the strip 12 < < 1). We will prove this Theorem in Sect. 2, after stating the results generalizing the two steps of Bagchi’s strategy for the zeta function. The proof of Bagchi’s Theorem for this family is an analogue of a proof for the Riemann zeta function that is simpler than Bagchi’s proof (it avoids both the use of the ergodic theorem and any tightness or weak-compactness argument).

1

We assume q > 17 to ensure this property; it also holds for q D 11.

Bagchi’s Theorem for Families of Automorphic Forms

183

In Sect. 5, we discuss very briefly how this strategy can in principle be applied to very general families of L-functions, as defined in [8].

Acknowledgements The simple proof of Bagchi’s Theorem for the Riemann zeta function, that we generalize here to modular forms, was found during a course on probabilistic number theory at ETH Zürich in the Fall Semester 2015; thanks are due to the students who attended this course for their interest and remarks, and to B. Löffel for assisting with the exercises (see [9] for a draft of the lecture notes for this course, especially Chapter 3). Thanks to the referee for carefully reading the text, and in particular for pointing out a number of confusing mistakes in references to the literature.

Notation As usual, jXj denotes the cardinality of a set. By f g for x 2 X, or f D O.g/ for x 2 X, where X is an arbitrary set on which f is defined, we mean synonymously that there exists a constant C > 0 such that jf .x/j 6 Cg.x/ for all x 2 X. The “implied constant” is any admissible value of C. It may depend on the set X which is always specified or clear in context. We write f g if f g and g f are both true. We use standard probabilistic terminology: a probability space .˝; ˙; P/ is a triple made of a set ˝ with a -algebra and a measure P on ˙ with P.˝/ D 1. We denote by E.X/ the expectation on ˝. The law of a random variable X is the measure  on the target space of X defined by .A/ D P.X 2 A/. If A ˝, then 1A is the characteristic function of A.

2 Equidistribution and Universality for Modular Forms in the Level Aspect We will prove Theorem 1 by combining the results of the following two steps, each of which will be proved in a forthcoming section. Throughout, we assume that q is a prime number > 17. Step 1. (Equidistribution; Bagchi’s Theorem) For q prime, we view the finite set S2 .q/ as a probability space with the probability measure proportional to the “harmonic” measure where f 2 S2 .q/ has weight 1 hf ; f i

184

E. Kowalski

in terms of the Petersson inner product. We write correspondingly Eq ./ or Pq ./ for the corresponding expectation and probability. Hence there exists a constant cq > 0 such that X cq Eq .'.f // D '.f / hf ; f i  f 2S2 .q/

for any 'W S2 .q/ ! C. From the Petersson formula, it is known that cq ! 1=.4/ as q ! C1 (see, e.g., Iwaniec and Kowalski [7, Chap. 14] or Cogdell and Michel [3]). Let D be a relatively compact open set in C such that D is invariant under complex conjugation. Define H.D/ to be the Banach space of functions holomorphic on D N with the norm and continuous and bounded on D, k'k1 D sup j'.s/j: N s2D

This is a separable complex Banach space. Define also HR .D/ to be the set of ' 2 H.D/ such that f .Ns/ D f .s/ for all s 2 D (this is well-defined since C is assumed to be invariant under conjugation). Note that the L-function of f , restricted to D, is an element of HR .D/ since the Hecke eigenvalues f .n/ are real for all n > 1. We define Lq to be the random variable S2 .q/ ! H.D/ mapping f 2 S2 .q/ to the restriction of L.f ; s/ to D. (This depends on D, but the choice of D will always be clear in the context.) N is a compact subset of the strip 1 < Re.s/ < 1, then we will show that If D 2 Lq converges in law to a random Dirichlet series. To define the limit, let .Xp /p be a sequence of independent random variables indexed by primes, taking values in the matrix group SU2 .C/ and distributed according to the probability Haar measure on SU2 .C/. N is a compact Theorem 2 (Bagchi’s Theorem for Modular Forms) Assume that D 1 subset of the strip 2 < Re.s/ < 1. Then, as q! C 1, the random variables Lq converge in law to the random Euler product LD .s/ D

Y p

det.1  Xp ps /1 D

Y

.1  Tr.Xp /ps C p2s /1

p

which is almost surely convergent in H.D/, and belongs almost surely to HR .D/. Step 2. (Support of the random Euler product) To deduce Theorem 1 from Theorem 2, we need the following computation of the support of the limiting measure. Theorem 3 Suppose that D is a disc with positive radius and diameter a segment N contained in 1 < Re.s/ < 1. The support of the law of the real axis, always with D 2 of the random Euler product LD contains the set of functions ' 2 H.D/ such that '.x/ > 0 for x 2 D \ R.

Bagchi’s Theorem for Families of Automorphic Forms

185

Note that since D\R is an interval of positive length in R, the condition '.x/ > 0 for all x 2 D \ R implies by analytic continuation that ' 2 HR .D/, which by Bagchi’s Theorem 2 is a necessary condition to be in the support of LD . Step 3. (Conclusion) The elementary Lemma 1 below, combined with Theorems 2 and 3, implies Theorem 1 in the form lim inf Pq .kL.f ; /  'k1 < "/ D lim inf q!C1

q!C1

X .q/

f 2S2 kL.f ;/'k1 0. We can easily deduce the “natural density” version from this: let A be the set of those f 2 S2 .q/ such that kL.f ; /  'k1 < "; then for any parameter  > 0, the definition of the harmonic measure on S2 .q/ gives    hf ; f i  X 1 hf ; f i  >  P 1 1 D E .f / .A/  P <  : q A q q jS2 .q/ j f 2A cq jS2 .q/ j cq jS2 .q/j There exists ı > 0 such that the first term is > ı > 0 for all q large enough by (2); on the other hand, a result of Cogdell and Michel [3, Corollary 1.16] and the classical relation between the Petersson norm and the symmetric square L-function at s D 1 (see, e.g., [7, (5.101)]) imply that we can find  > 0 such that lim Pq

q!C1



 ı hf ; f i <  < : cq jS2 .q/j 2

For this value of , we obtain lim inf q!C1

X 1 ı > 0: 1>  jS2 .q/ j f 2A 2

More precisely, the result of Cogdell–Michel is that for any  > 0, we have lim Pq .L.Sym2 f ; 1/ 6 / D F.log /

q!C1

where F is the limiting distribution function for the special value at 1 of the symmetric square L-function of f 2 S2 .q/ . Since F.x/ ! 0 when x ! 1, we obtain the result. Remark 1 It would also be possible to argue throughout with the uniform probability measure on S2 .q/ ; the only change would be a slightly different form of Theorem 2, where the random variables .Xp / would not be identically distributed (compare with the equidistribution theorems of Serre [16] and Conrey et al. [4]).

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Lemma 1 Let M be a separable complete metric space and .Xn / a sequence of random variables with values in M that converge in law to X. Let S be the support of the law of X. Then for any x 2 S, and any open neighborhood U of x, we have lim inf P.Xn 2 U/ > 0: n!C1

Proof By classical criteria for convergence in law, we have lim inf P.Xn 2 U/ > P.X 2 U/ n!C1

(3)

for any open set U X (see, e.g., [2, Theorem 2.1 (iv)]). Since x 2 S, we have P.X 2 U/ > 0, hence the result. t u

3 Proof of Theorem 2 We begin with some preliminaries concerning the random Euler product LD . In fact, if will be convenient to view it as a holomorphic function on larger domains then D, in a way that will be clear below. For this purpose, we fix a real number 0 such N is contained in the half-plane S that 12 < 0 < 1, and such that the compact set D defined by Re.s/ > 0 . We recall that for  > 0, the d-th Chebyshev polynomial is defined by U .2 cos.x// D

sin.. C 1/x/ : sin.x/

The importance of these polynomials for us lies in their relation with the representation theory of SU2 .C/, namely !  ix e 0 U .2 cos.x// D Tr Sym 0 eix for any x 2 R, where Symd is the d-th symmetric power of the standard twodimensional representation of SU2 .C/. We define a sequence of random variables .Yn /n>1 by Y Yn D U .Tr.Xp //: p jjn

In particular, we have Yn Ym D Ynm if n and m are coprime, and Yp D Tr.Xp / if p is prime. The sequence .Yp / is independent and Sato–Tate distributed. Moreover, since jU .t/j 6  C 1 for all  > 0 and all t 2 R, we have Y jYn j 6 . C 1/ D d.n/ n" p jjn

for n > 1 and " > 0, where the implied constant depends only on ".

Bagchi’s Theorem for Families of Automorphic Forms

187

Lemma 2 (1) Almost surely, the random Euler product Y

det.1  Xp ps /1

p

converges and defines a holomorphic function on S. In particular, LD converges almost surely to define an H.D/-valued random variable. (2) Almost surely, we have Y

X

det.1  Xp ps /1 D

p

Yn ns

n>1

for all s 2 S, and in particular LD coincides with the random Dirichlet series on the right. (3) For > 1=2 and u > 2, we have ˇ2  ˇX ˇ ˇ E ˇ Yn n ˇ 1; n6u

where the implied constant depends only on . Proof (1) Let be a fixed real number such that can write

1 2

< < 0 . By expanding, we

 log det.1  Xp ps / D Yp ps C gp .s/ where the random series X

gp .s/

p

converges absolutely (and surely) for Re.s/ > 1=2. Since E.Yp p / D 0 and E.Yp2 p2 / D p2 , Kolmogorov’s Three Series Theorem (see, e.g., [14, Theorem 0.III.2]) implies that the random series X

Yp p

p

converges almost surely. By well-known results on Dirichlet series, this means that the random series X Yp ps p

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E. Kowalski

converges almost surely to a holomorphic function on the half-plane S. This implies the first statement by taking the exponential. The second follows by restricting to D N is contained in S. since D (2) We first show that, almost surely, the random Dirichlet series X

Q L.s/ D

Yn ns

n>1

converges and defines a function holomorphic on S. The key point is that the variables Yn for n squarefree form an orthonormal system: we have E.Yn Ym / D ı.n; m/ if n and m are squarefree numbers. Indeed, if n 6D m, there is a prime p dividing only one of n and m, say p j n, and then by independence we get E.Yn Ym / D E.Yp /E.Yn=p Ym / D 0; and if n D m is squarefree, then we have E.Yn2 / D

Y

E.Yp2 / D 1:

pjn

Fix again such that 12 < < 0 . By the Rademacher–Menchov Theorem (see, e.g., [9, Theorem B.8.4]), the random series X[ n

Yn n

over squarefree numbers converges almost surely. By elementary factorization and properties of products of Dirichlet series (the product of an absolutely convergent Dirichlet series and a convergent one is convergent, see, e.g., [6, Theorem 54]) the same holds for X Yn n : n>1

Q As in (1), this gives the almost sure convergence of the series defining L.s/ to a holomorphic function in S. Restricting gives the H.D/-valued random variable LQ D . Finally, almost surely both the random Euler product and the random Dirichlet series converge and are holomorphic for Re.s/ > 0 . For Re.s/ > 3=2, they converge absolutely, and coincide by a well-known formal Euler product computation: for any prime p and any x 2 R, denoting t.x/ D

  ix e 0 0 eix

Bagchi’s Theorem for Families of Automorphic Forms

189

we have det.1  t.x/ps /1 D .1  eix ps /1 .1  eix ps /1 X X D Tr Sym .t.x//ps D U .x/ps >0

>0

(compare the discussion of Cogdell and Michel in [3, Sect. 2]). By analytic continuation, we deduce that LD D LQ D almost surely as H.D/-valued random variables. (3) Since the random variables Yn are real-valued, we have ˇ2  ˇX X ˇ ˇ Yn n ˇ D E ˇ n6u

1 E.Yn Ym /:

.nm/ n;m6u

For given n and m, let d D .n; m/ and n0 D n=d, m0 D m=d. Then by multiplicativity and independence of the variables .Yp /p , we have E.Yn Ym / D E.Yd2 /E.Yn0 /E.Ym0 /: By the definition of Yp , we have E.Yn0 / D 0 if n0 is divisible by a prime p with odd exponent, and similarly for E.Ym0 /. Hence we have E.Yn Ym / D 0 unless both n0 and m0 are squares. Therefore ˇ2  X E.Y 2 / X ˇX 1 ˇ ˇ d E ˇ Yn n ˇ 6 E.Ym Yn / < C1 2

2

d .mn/ n6u d>1 m;n>1 since > 1=2 and E.Yn / n" for any " > 0. t u The key arithmetic properties of the family S2 .q/ of modular forms that are required in the proof of Theorem 2 are the following: Proposition 1 (Local Spectral Equidistribution) As q ! C1, the sequence . f .p//p of Fourier coefficients of f 2 S2 .q/ converges in law to the sequence .Yp /p . Proof This is a well-known consequence of the Petersson formula, see, e.g., [11, Proposition 8], [12, Appendix] or [3, Proposition 1.9]; here restricting to prime level q and weight 2 also simplifies matters since this ensures that the old space of S2 .q/ is zero. t u Proposition 2 (First Moment Estimate) There exists an absolute constant A > 1 such that for any real number ı > 0 with ı < 1=2, and for any s 2 C such that 1 C ı 6 Re.s/, we have 2 Eq .jL.f ; s/j/ .1 C jsj/A ; where the implied constant depends only on ı.

190

E. Kowalski

Proof This follows easily, using the Cauchy–Schwarz inequality, from the secondmoment estimate [10, Proposition 5] of Kowalski and Michel (with  D 0); although this statement is not formally the same, it is in fact a more difficult average (it operates closer to the critical line). t u We now prove some additional lemmas. Lemma 3 (Polynomial Growth) For any real number > 0 , we have ˇ ˇX ˇ ˇ E ˇ Yn ns ˇ 1 C jsj n>1

uniformly for all s such that Re.s/ > > 0 . Proof We write L.s/ D

X

Yn ns :

n>1

This is almost surely a function holomorphic on the half-plane S. The series X Yn n 0 n>1 converges almost surely. Therefore the partial sums Su D

X Yn n 0 n6u

are bounded almost surely. By summation by parts, it follows from the convergence of the series L.s/ that for any s with real part Re.s/ > > 0 , we have Z

C1

L.s/ D .s  0 / 1

Su du; s

u 0 C1

where the integral converges almost surely. Hence almost surely Z

C1

jL.s/j 6 .1 C jsj/ 1

jSu j du:



u 1 C1

Fubini’s Theorem and the Cauchy–Schwarz inequality then imply Z

C1

E.jL.s/j/ 6 .1 C jsj/

E.jSu j/ 1

Z 6 .1 C jsj/

1

by Lemma 2 (3).

du u  0 C1

C1

E.jSu j2 /1=2

du

1 C jsj u  0 C1 t u

Bagchi’s Theorem for Families of Automorphic Forms

191

We now consider some elementary approximation statements of the L-functions and of the random Dirichlet series by smoothed partial sums. For this, we fix once and for all a smooth function 'W Œ0; C1Œ! Œ0; 1 with compact support such that '.0/ D 1, and we denote 'O its Mellin transform. We also fix T > 1 and a compact interval I in 1=2; 1Œ such that the compact rectangle R D I  ŒT; T C is contained in S and contains D in its interior. We then finally define ı > 0 so that minfRe.s/ j s 2 Rg D

1 C 2ı: 2

Lemma 4 For N > 1, define the H.D/-valued random variable .N/

LD D

X

Yn '

n>1

n ns : N

We then have .N/

E.kLD  LD k1 / N ı for N > 1, where the implied constant depends on D. Proof We again write L.s/ D

X

Yn ns

n>1

when we wish to view the Dirichlet series as defined and holomorphic (almost surely) on S. For any s in the rectangle R, we have almost surely the representation L.s/  L.N/ .s/ D 

1 2i

Z

w L.s C w/'.w/N O dw

(4)

.ı/

by standard contour integration.2 We also have almost surely for any v in D the Cauchy formula .N/

LD .v/  LD .v/ D

1 2i

Z

.L.s/  L.N/ .s// @R

ds ; sv

2 Here and below, it is important that the “almost surely” property holds for all s, which is the case because we work with random holomorphic functions, and not with particular evaluations of these random functions at specific points s.

192

E. Kowalski

where the boundary of R is oriented counterclockwise. The definition of the rectangle R ensures that js  vj1  1 for v 2 D and s 2 @R, and therefore Z .N/ jL.s/  L.N/ .s/j jdsj: kLD  LD k1

@R

Using (4) and writing w D ı C iu with u 2 R, we obtain Z Z .N/ jL.ı C iu C s/j j'.ı O C iu/jjdsjdu: kLD  LD k1 N ı @R

R

Therefore, taking the expectation, and using Fubini’s Theorem, we get Z Z   .N/ ı E jL.ı C iu C s/j j'.ı O C iu/jjdsjdu E.kLD  LD k1 / N @R

N ı

R

Z

  E jL.ı C iu C C it/j j'.ı O C iu/jdu:

sup sD Cit2R R

We therefore need to bound Z   E jL.ı C iu C C it/j j'.ı O C iu/jdu: R

for some fixed C it in the compact rectangle R. The real part of the argument ı C iu C C it is  ı > 12 C ı by definition of ı, and hence E.jL.ı C iu C C it/j/ 1 C j  ı C iu C C itj 1 C juj uniformly for C it 2 R and u 2 R by Lemma 3. Since 'O decays faster than any polynomial at infinity, we conclude that Z

  E jL.ı C iu C C it/j j'.ı O C iu/jdu 1 R

uniformly for s D C it 2 R, and the result follows. We proceed similarly for the L-functions.

t u

Lemma 5 For N > 1 and f 2 S2 .q/ , define L.N/ .f ; s/ D

X n>1

f .n/'

n N

ns ;

.N/

and define Lq to be the H.D/-valued random variable mapping f to L.N/ .f ; s/ restricted to D. We then have ı Eq .kLq  L.N/ q k1 / N

for N > 1 and all q.

Bagchi’s Theorem for Families of Automorphic Forms

193

Proof For any s 2 R, we have the representation L.f ; s/  L.N/ .f ; s/ D 

1 2i

Z

w L.f ; s C w/'.w/N O dw:

(5)

.ı/

and for any v with Re.v/ > 1=2, Cauchy’s theorem gives L.f ; v/  L

.N/

Z

1 .f ; v/ D 2i

.L.f ; s/  L.N/ .f ; s// @R

ds ; sv

where the boundary of R is oriented counterclockwise. As in the previous argument, we deduce that Z .N/ kLq  Lq k1

jL.f ; s/  L.N/ .f ; s/jjdsj @R

for f 2 S2 .q/ . Taking the expectation with respect to f and changing the order of summation and integration leads to Z    

k Eq jL.f ; s/  L.N/ .f ; s/j jdsj Eq kLq  L.N/ 1 q @R

 

sup Eq jL.f ; s/  L.N/ .f ; s/j :

(6)

s2@R

Applying (5) and using again Fubini’s Theorem, we obtain   Eq jL.f ; s/  L.N/ .f ; s/j N ı

Z

  j'.ı O C iu/jEq jL.f ; ı C iu C C it/j du R

for s 2 @R. Since  ı >

1 2

C ı, we get

  Eq jL.f ; ı C iu C C it/j .1 C juj/A

(7)

by Proposition 2, where A is an absolute constant. Hence   Eq jL.f ; s/  L.N/ .f ; s/j N ı

Z

j'.ı O C iu/j.1 C juj/A du N ı :

(8)

R

We conclude from (6) that   ı Eq kLq  L.N/ q k1 N ; as claimed.

t u

194

E. Kowalski

Proof (Proof of Theorem 2) A simple consequence of the definition of convergence in law shows that it is enough to prove that for any bounded and Lipschitz function f W H.D/!C, we have Eq .f .Lq //!E.f .LD // as q! C 1 (see [2, p. 16, (ii)) (iii) and (1.1), p. 8]). To prove this, we use the Dirichlet series expansion of LD given by Lemma 2 (2). Let N > 1 be some integer to be chosen later. Let L.N/ q D

X

f .n/ns '

n

n>1

N

(viewed as random variable defined on S2 .q/ ) and LN D

X n>1

Yn ns '

n N

be the smoothed partial sums of the Dirichlet series, as in Lemmas 5 and 4. We then write jEq .f .Lq //  E.f .L//j 6 jEq .f .Lq /  f .L.N/ q //jC .N/ //j C jE.f .L.N/ /  f .L//j: jEq .f .L.N/ q //  E.f .L

Since f is a Lipschitz function on H.D/, there exists a constant C > 0 such that jf .x/  f .y/j 6 Ckx  yk1 for all x, y 2 H.D/. Hence we have jEq .f .Lq //  E.f .L//j 6 CEq .kLq  L.N/ q k1 /C .N/ //j C CE.kL.N/  Lk1 /: jEq .f .L.N/ q //  E.f .L

Fix " > 0. Lemmas 5 and 4 together show that there exists some N > 1 such that Eq .kLq  L.N/ q k1 / < " for all q > 2 and E.kL.N/  Lk1 / < ":

Bagchi’s Theorem for Families of Automorphic Forms

195

We fix such a value of N. By Proposition 1 (and composition with a continuous .N/ function), the random variables Lq (which are Dirichlet polynomials) converge in .N/ law to L as q! C 1. We deduce that we have jEq .f .Lq //  E.f .L//j < 4" for all q large enough. This finishes the proof.

t u

4 Proof of Theorem 3 For the computation of the support of the random Dirichlet series L.s/, we apply a trick to exploit the analogous result known for the case of the Riemann zeta function. c 2 the product of copies of the unit circle indexed by primes, so an We denote SU c 2 is a family of matrices in SU2 .C/ indexed by p. element .xp / of SU The assumptions on D in Theorem 33 imply that there exists  such that 1=2 <  < 1 and r > 0 such that D D fs 2 C j js  j 6 rg fs 2 C j 1=2 < Re.s/ < 1g: Lemma 6 Let N be an arbitrary positive real number. The set of all series X Tr.xp / p>N

ps

;

c2 .xp / 2 SU

which converge in H.D/ is dense in the subspace HR .D/. In the proof and the next, we allow ourselves the luxury of writing sometimes k'.s/k1 instead of k'k1 . Proof Bagchi [1, Lemma 5.2.10] proves (using results of complex analysis due to Bernstein, Polyá, and others) that the set of series X ei p p>N

ps

;

p 2 R

that converge in H.D/ is dense in H.D/ (precisely, he proves this for N D 1, but the same proof applies to any value of N). If ' 2 HR .D/ and " > 0, we can therefore find real numbers . p / such that  '.s/ X ei p  "      < : s 1 2 p 2 p>N 3

These assumptions could be easily weakened, as has been done for Voronin’s Theorem.

196

E. Kowalski

It follows then that  '.Ns/ X ei p  "    ; < ;  s 1 2 p 2 p>N hence (since ' 2 HR .D/) that  X ei p C ei p    '.s/   < "; s 1 p p>N which gives the result since i p

e

i p

Ce



ei p 0 D Tr 0 ei p



is the trace of a matrix in SU2 .C/. We will use this to prove:

t u

Proposition 3 The support of the law of log LD .s/ D 

X

log det.1  Xp ps /

p

in H.D/ is HR .D/. Proof Since Xp 2 SU2 .C/, the function log L.s/ is almost surely in the space HR .D/. Since the summands are independent, a well-known result concerning the support of random series (see, e.g., [9, Proposition B.8.7]) shows that it suffices to prove that the set of convergent series 

X

log det.1  xp ps /;

c 2; .xp / 2 SU

p

is dense in HR .D/. Denote L.sI .xp // this series, when it converges in H.D/. We can write 

X

log det.1  xp ps / D

p

X Tr.xp / p

ps

C g.sI .xp //

where s 7! g.sI .xp // is holomorphic in the region Re.s/ > 1=2. Indeed g.sI .xp // D

X p

  X log 1 C Tr.xp /k p.kC2/s : k>0

Bagchi’s Theorem for Families of Automorphic Forms

197

Fix ' 2 HR .D/ and let " > 0 be fixed. There exists N > 1 such that   X  X   log 1 C Tr.xp /k p.kC2/s  s 7! p>N

k>0

1

N The left-hand side is the norm of log L.sI .xp //  g.sI .xp // 

X Tr.xp / p6N

ps

 '1 .s/ D log L.sI .xp //  '.s/  g.xI .xp //;

and by (9), we obtain k log L.sI .xp //  '.s/k1 < 2": This implies the lemma. t u Using composition with the exponential function and a lemma of Hurwitz (see, e.g., [17, 3.45]) on zeros of limits of holomorphic functions, we see that the support of the limiting Dirichlet series LD in H.D/ is the union of the zero function and the set of functions ' 2 HR .D/ such that '. / > 0 for 2 D \ R. In particular, this proves Theorem 3.

5 Generalizations It is clear from the proof that Bagchi’s Theorem should hold in considerable generality for any family of L-functions. Indeed, the crucial ingredients are the local spectral equidistribution (Proposition 1) and the first moment estimate (Proposition 2). The first result is a qualitative statement that is understood to be at the core of any definition of “family” of L-functions (this is explained in [8], but also appears, with a different terminology, for the families of Conrey et al. [5] and Sarnak et al. [15]); it is now know in many circumstances (indeed, often in quantitative form).

198

E. Kowalski

The moment estimate is typically derived from a second-moment bound, and is also definitely expected to hold for a reasonable family of L-functions, but it has only been proved in much more restricted circumstances than local spectral equidistribution. However, it is very often the case that one can at least prove (using local spectral equidistribution) a weaker statement: for some 1 such that 1=2 < 1 < 1, the second moment of the L-functions satisfies the analogue of Proposition 2; an analogue of Bagchi’s Theorem then follows at least for compact discs in the region 1 < Re.s/ < 1. As far as universality (i.e., Theorem 3) is concerned, one may expect that (using tricks similar to the proof of Theorem 3) only two different cases really occur, depending on whether the coefficients of the L-functions are real (as in our case) or complex (as in the case of vertical translates of a fixed L-function). Acknowledgements This work was partially supported by a DFG-SNF lead agency program grant (grant 200021L_153647).

References 1. B. Bagchi, Statistical behaviour and universality properties of the Riemann zeta function and other allied Dirichlet series, PhD thesis, Indian Statistical Institute, Kolkata, 1981; available at library.isical.ac.in/jspui/handle/10263/4256 2. P. Billingsley, Convergence of Probability Measures (Wiley, New York, 1968) 3. J. Cogdell, P. Michel, On the complex moments of symmetric power L-functions at s D 1. Int. Math. Res. Not. 31, 1561–1617 (2004) 4. B. Conrey, W. Duke, D. Farmer, The distribution of the eigenvalues of Hecke operators. Acta Arith. 78, 405–409 (1997) 5. J.B. Conrey, D. Farmer, J. Keating, M. Rubinstein, N. Snaith, Integral moments of L-functions. Proc. Lond. Math. Soc. 91 33–104 (2005) 6. G.H. Hardy, M. Riesz, The General Theory of Dirichlet’s Series. Cambridge Tracts in Mathematics, vol. 18 (Cambridge University Press, Cambridge, 1915) 7. H. Iwaniec, E. Kowalski, Analytic Number Theory. Colloquium Publications, vol. 53 (American Mathematical Society, Providence, 2004) 8. E. Kowalski, Families of Cusp Forms (Pub. Math. Besançon, 2013), pp. 5–40 9. E. Kowalski, Arithmetic Randonnée: An Introduction to Probabilistic Number Theory. ETH Zürich Lecture Notes, www.math.ethz.ch/~kowalski/probabilistic-number-theory.pdf 10. E. Kowalski, Ph. Michel, The analytic rank of J0 .q/ and zeros of automorphic L-functions. Duke Math. J. 100, 503–542 (1999) 11. E. Kowalski, Y.-K. Lau, K. Soundararajan, J. Wu, On modular signs. Math. Proc. Camb. Philos. Soc. 149, 389–411 (2010). doi:10.1017/S030500411000040X 12. E. Kowalski, A. Saha, J. Tsimerman, Local spectral equidistribution for Siegel modular forms and applications. Compos. Math. 148, 335–384 (2012) 13. A. Laurinˇcikas, K. Matsumoto, The universality of zeta-functions attached to certain cusp forms. Acta Artih. 98, 345–359 (2001) 14. D. Li, H. Queffélec, Introduction à l’étude des espaces de Banach; Analyse et probabilités, Cours Spécialisés 12, S.M.F, 2004 15. P. Sarnak, S.-W. Shin, N. Templier, Families of L-functions and their symmetry, in Families of automorphic forms and the trace formula, Simons Symposia (Springer, New York, 2016), pp. 531–578

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16. J-P. Serre, Répartition asymptotique des valeurs propres de l’opérateur de Hecke Tp . J. Am. Math. Soc. 10, 75–102 (1997) 17. E.C. Titchmarsh, The Theory of Functions, 2nd edn. (Oxford University Press, Oxford, 1939) 18. S.M. Voronin, Theorem on the ‘universality’ of the Riemann zeta function, Izv. Akad. Nauk SSSR 39, 475–486 (1975); translation in Math. USSR Izv. 9, 443–445 (1975)

The Liouville Function and the Riemann Hypothesis Michael J. Mossinghoff and Timothy S. Trudgian

Abstract For nearly a century, mathematicians have explored connections between the Liouville function and the Riemann hypothesis. We describe a number of connections P regarding oscillations in sums involving the Liouville function, including P L0 .x/ D nx .n/, first studied by Pólya, and L1 .x/ D nx .n/=n, explored by Turán. We establish new lower bounds p on the size of the oscillations in such sums. p In particular, we prove that L0 .x/ > x infinitely often, show that L1 .x/ < 1= x infinitely often, and investigate sign Pchanges in a family of functions that interpolate these functions, namely L˛ .x/ D nx .n/=n˛ , where ˛ 2 Œ0; 1.

1 Introduction The Liouville function .n/ is the completely multiplicative arithmetic function whose value is 1 at each prime, so .n/ D .1/˝.n/ , where ˝.n/ is the number of prime factors of n, counting multiplicity. For nearly 100 years mathematicians have explored connections between this function and the Riemann hypothesis. Ultimately, these are connected by the Dirichlet series for the Liouville function: for 1, using the Euler product representation of the zeta function we obtain Q 2s 1 Y Y X .1/k X .n/ / .2s/ p .1  p D Q D .1 C ps /1 D D : s 1 ks .s/ p ns p .1  p / p p k0 n1 Here and throughout this article, s denotes a complex variable with real part and imaginary part t.

M.J. Mossinghoff () Department of Mathematics and Computer Science, Davidson College, Davidson, NC 28035-6996, USA e-mail: [email protected] T.S. Trudgian School of Physical, Environmental and Mathematical Sciences, The University of New South Wales Canberra, Canberra, ACT 2610, Australia e-mail: [email protected] © Springer International Publishing AG 2017 H. Montgomery et al. (eds.), Exploring the Riemann Zeta Function, DOI 10.1007/978-3-319-59969-4_9

201

202

M.J. Mossinghoff and T.S. Trudgian

For a nonnegative real number ˛, let L˛ .x/ denote the following weighted sum: L˛ .x/ D

X .n/ nx

n˛

:

(1)

The two special cases ˛ D 0 and ˛ D 1 have a long history in analytic number theory, in topics surrounding the Riemann hypothesis. Pólya investigated the sum L0 .x/ [which often appears in the literature as L.x/] in a 1919 publication [27]. His main result was a proof that if p > 7 is a prime with p p  3 mod 4, such that the quadratic imaginary number field Q. p/ has class number 1, then L0 .x/ must vanish at x D .p  3/=4. He then reported computing L0 .x/ up to approximately 1500, and noted that it was never positive over this range, after the trivial value L0 .1/ D 1. Pólya wrote: I mention this observation, in order to prompt perhaps further numerical research. The proof of (1) [referring to L0 .n/  0], even only for sufficiently large n, would produce a proof of the Riemann hypothesis, after known properties of the -function and a function-theoretic result of Mr. Landau.

The statement that L0 .n/  0 for sufficiently large n is often described in the literature as Pólya’s conjecture, but in fact Pólya only noted this implication, and did not conjecture its validity, at least in print. We therefore employ the term Pólya’s problem in this article for the question of sign changes, and more broadly of oscillations, in the function L0 .x/. Turán studied the weighted sum L1 .x/ [which often appears in the literature as T.x/] in 1948 [31],P in connection with the zeros of the truncated series for the zeta s function, Ux .s/ D nx 1=n p . Turán proved that if there exists a positive constant K such that L1 .x/ > K= x for p sufficiently large x, then Ux .s/ does not vanish in the half plane  1 C K= x, which would imply the Riemann hypothesis. (Somewhat weaker conditions suffice as well; see [31, 32].) Turán reported that a coterie of Danish colleagues computed L1 .x/ for x  1000, and found that this function remained positive throughout this range. Perhaps because of this, in later literature the statement that L1 .x/ is never negative is sometimes stated as Turán’s conjecture. However, in a later paper [32] Turán took issue with the term: Strangely, this [referring to the positivity of L1 .n/] is described in subsequent work as “Turán’s conjecture,” even claimed for n  1, but this occurs nowhere in [31], for n  1 not even implicitly claimed.

We use the term Turán’s problem in this article to refer to the question of sign changes and oscillations in L1 .x/. Turán’s original motivation for studying this problem was settled by Montgomery in 1983 [22], who proved that Ux .n/ does have zeros some distance to the right of the D 1 line in the complex plane. He proved that for every positive constant c < 4  1 and every integer x > x0 .c/ the function Ux .s/ has a zero in the half plane

> 1 C c log log x= log x. A good deal is known about both Pólya’s and Turán’s problem. In 1958, Haselgrove [11] proved that both L0 .x/ and L1 .x/ change sign infinitely often.

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Lehman [19] first determined an explicit location where L0 .x/ is positive, proving in 1960 that L0 .906 180 359/ D 1. The location of the first sign crossing of L0 .x/ for x  2 was not found until 1980, when Tanaka [30] established that this occurs at x D 906 150 257. In Turán’s problem, the first sign crossing was found more recently by Borwein, Ferguson, and the first author [5]. It occurs at x D 72 185 376 951 205. It is well known (and was certainly known to Pólya and Turán) that the Riemann hypothesis would follow if one could bound the oscillations in L0 .x/ or L1 .x/. The Riemann hypothesis, as well as the simplicity of the zeros of .s/, would follow if there exists a positive constant C such that any one of the following inequalities holds for sufficiently large x: p p L0 .x/ L0 .x/ p < C; p > C; L1 .x/ x < C; L1 .x/ x > C: x x

(2)

This follows from Landau’s theorem regarding singularities of the Laplace transform of a nonnegative function (see, for p instance, [23, Lemma 15.1]). It is easy to see that bounding L0 .x/= x on both sides would yield the Riemann hypothesis. Starting with the Dirichlet series for the Liouville function in the complex variable s D C it, one may compute   X .n/ X L0 .n/  L0 .n  1/ X .2s/ 1 1 D D D L .n/  0 .s/ ns ns ns .n C 1/s n1 n1 n1 D

X n1

Z

nC1

L0 .n/ n

s xsC1

Z

1

dx D s 1

L0 .x/ dx; xsC1

(3) p so that clearly L0 .x/ D O. x/ implies that the right side of (3) is analytic on > 1=2, and therefore so is the left side, which would establish the Riemann hypothesis. One can deduce that the zeros of the zeta function on the critical line are all simple in this case as well. A result of Ingham [14] in 1942 cast some doubt on the possibility of proving p the Riemann hypothesis and the simplicity of the zeros of .s/ by bounding L0 .x/= x or similar functions. Ingham proved that such a result would imply considerably more: it would follow that infinitely many integer relations exist among the ordinates of the zeros of .s/ on the critical line in the upper half plane. Since there seems to be no other particular reason to suspect the existence of such relations, and no evidence of such relations has been detected, it seems plausible that these (suitably scaled) sums involving the Liouville function may be unbounded. In this article, we focus on obtaining new and larger bounds on these oscillations. Some prior bounds on oscillations for L0 .x/ are known. Anderson and Stark [2] provedp that single values of L0 .x/ could be employed to quantify the oscillations of L0 .x/= x. They showed that L0 .x/ L0 .x0 /  I.x0 / lim sup p  ; p x0 x x!1

(4)

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M.J. Mossinghoff and T.S. Trudgian p 2 2.3=2/ p 3.3/ x

p  6:437= x. Lehman [19] reported p that L0 .906 400 000/ D 708, whence by (4), L0 .x/ > 0:0234833 x infinitely often. Tanaka [30] found that L0 .x/ reaches a maximum value of 829 for x  109 , and while he did not report that this value occurred p at x D 906 316 571, with this datum (4) shows that L0 .x/ > 0:0275036 x infinitely often. The more recent work of Borwein, Ferguson, and the first author [5] determined that L0 .351 753 358 289 465/ D 1;160;327, which establishes that where I.x/ satisfies jI.x/  1j 

p L0 .x/ > 0:0618672 x infinitely often. Further, as noted by Humphries [13], the method of Anderson p p and Stark also shows that lim infx!1 L0 .x/= x  .L0 .x0 /  I.x0 //= x0 for any particular x0 . The data point L0 .72 204 113 780 255/ D 11 805 117 from [5] establishes that p L0 .x/ < 1:389278 x infinitely often. In this article, we prove even larger oscillations in L0 .x/. Due to some symmetry in these problems, our results apply equally to L1 .x/, after suitable scaling. We prove the following result in Sect. 6. Theorem 1 Each of the following inequalities is satisfied for infinitely many integers x: p L0 .x/ > 1:0028 x; p L1 .x/ < 1:0028= x;

p L0 .x/ < 2:3723 x; p L1 .x/ > 2:3723= x:

The more general family L˛ .x/ with ˛ 2 Œ0; 1 was studied by the authors [24], by Humphries [13], and by Akbary et al. [1]. It will be seen in Sect. 2 that if L˛ .x/ eventually has no sign changes for some ˛ 2 Œ0; 1=2/, or if L˛ .x/  .2˛/=.˛/ eventually has no sign changes for some ˛ 2 .1=2; 1, then the Riemann hypothesis follows, as well as the simplicity of the zeros, and the existence of infinitely many integer relations among the ordinates of the zeros of the zeta function in the upper half plane. Because of this, in [24] the authors raised the question of whether these functions do change sign infinitely often. Our methods in this article allow us to resolve this question for almost 60% of the interval Œ0; 1. We establish the following theorem in Sect. 6. Theorem 2 For each ˛ 2 Œ0; 0:29714 : : :/, the function L˛ .x/ exhibits infinitely many sign changes, and the same conclusion holds for each ˛ 2 .0:70285 : : : ; 1 for the function L˛ .x/  .2˛/=.˛/. Some additional explicit results regarding the oscillations of L˛ .x/ for particular values of ˛ appear in Sect. 6 (see Theorem 8).

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This article is organized in the following way. Section 2 describes a normalization of the functions L˛ .x/ that allows us to treat their oscillations in a uniform way. Section 3 is an interlude on sums involving the Möbius function. Section 4 reports on a method involving the so-called weak independence of a set of real numbers, which quantifies oscillations in sums of arithmetic functions. Section 5 then describes a more recent strategy developed by Best and the second author [4] to obtain improved bounds on oscillations for sums involving the Möbius function. Section 6 describes an adaptation of this method to sums involving the Liouville function, and establishes Theorems 1 and 2. Finally, Sect. 7 briefly discusses the potential for future improvements.

2 Generalizing the Problems of Pólya and Turán We begin by defining some functions that will allow a more uniform treatment of oscillation questions for the functions L˛ .x/ with ˛ 2 Œ0; 1. In [24], it is shown that the Riemann hypothesis is true if and only if L˛ .x/ converges to .2˛/=.˛/ for each ˛ 2 .1=2; 1. In fact, convergence at a particular ˛ in this range is tantamount to showing that the zeta function has no zeros in the half plane  ˛, so in particular the fact that L1 .x/ converges to 0 is equivalent to the Prime Number Theorem. Moreover, it is shown in [24] that the Riemann hypothesis produces an 1 error bound of O.x 2 ˛C / for the quantity L˛ .x/  .2˛/=.˛/. A similar statement is shown to be equivalent to the Riemann hypothesis for the case of small ˛: for each 1 ˛ 2 Œ0; 1=2/, this conjecture is equivalent to the statement that L˛ .x/ D O.x 2 ˛C /. We describe the special case ˛ D 1=2 shortly, but following [24] we define 8 ˆ L˛ .x/; ˆ ˆ ˆ ˆ 1=2 and converges in > 0 on the Riemann hypothesis. The following formula is derived there: 8 ˆ f˛ .s/; if 0  ˛ < 1=2 or ˛ D 1; ˆ ˆ ˆ ˆ 1 C, for some fixed positive constant C and all sufficiently large x, would also imply the existence of infinitely many integer linear relations among the ordinates of the zeros of the zeta function in the upper half plane. In this case, the function corresponding to F0 .s/ from (8) is F.s/ D 1=.s C 1=2/.s C 1=2/. This has no pole at the origin, and the residue at i , where  D 1=2 C i is a zero of .s/, is Res.F; i / D 1= 0 ./. The lack of pole at the origin for F.s/ produces qualitatively different behavior for the Möbius sums, compared to the functions L˛ .x/: the sums M˛ .x/ exhibit no sign bias. We remark that Odlyzko and te Riele’s disproof of the Mertens conjecture employed the analog of Theorem 4 for M0 .x/. They showed that the sum B .u/ D 2
T:

(18)

Our experiments with the choice of kernel consisted of evaluating the limiting values of the bounds in (13) and (14) as N ! 1 using these two possibilities for kT .t/, for a number of sets  0 obtained from various choices of T and n. We found that the function (18) consistently produced better results, just as Best and the second author concluded in [4, Fig. 2]. It is possible that another choice of kernel could produce better bounds, but Odlyzko and te Riele [25, Sect. 4.1] indicate that potential improvements from such a change are rather limited. With our kernel selected, we have four additional parameters to choose for our computation: • T, the height on the critical line for the zeros of .s/, or equivalently m, the number of zeros in  \ Œ0; T. • n, the size of the set  0 , and the dimension of the lattice 0 . • b, the number of bits of precision for computing the zeros of .s/. • ı, the parameter governing the quality of the basis produced by LLL, as well as its run time. In the main computation of [4], the authors selected n to be rather large, at 500, and b very large as well, at nearly 30;000. This makes for a time-consuming lattice reduction step, but they compensated in part for this by selecting a modest value for ı at 0:3, and picking m somewhat small at 2000. This computation employed 1501 lattices, each of dimension 500 or 501, and produced a value of N close to 5000. We take a different approach in this computation. We wish to show that A0 .u/ > 1 infinitely often: this can be computed with a number of different choices of the parameters. If we select n to be small, then we need to check many zeros of .s/ before finding enough with sufficiently large residues, or we need to select T large so that the weight function is more generous at the zeros we pick. In any event this strategy produces many lattices to analyze. Choosing smaller values for n would also encourage selecting a larger ı, which would produce a better result. On the other hand, if we pick n to be large, then we will not need as many zeros, so we will have fewer lattices to reduce, but each one will require more time, and we may need to pick ı smaller. The LLL algorithm has worst-case running time O.n6 b3 /, but often performs better in practice. Through experiments, we found that b needed to grow approximately linearly in n, and some empirical trials using b D .n/ found that the total run time required by running LLL followed by Gram–Schmidt grew approximately as n4 .

The Liouville Function and the Riemann Hypothesis Table 1 Timing estimates (in core-days) for establishing that A0 .u/ > 1 infinitely often

217 n 240 250 260 270 280 290 300

m 5000 3700 3000 2600 2400 2200 2000

b 6500 7000 7500 8000 8500 9000 9500

C 1.0033 1.0034 1.0023 1.0023 1.0056 1.0065 1.0042

Time 417 376 407 435 471 476 480

1–89, 91–94, 96–98, 100–108, 111, 113–119, 121, 123–130, 132, 134–136, 138, 140–142, 145– 147, 149, 151–153, 156–160, 163, 164, 169, 171, 172, 174, 175, 178, 180–182, 186, 187, 189, 192, 193, 197, 199, 204, 205, 208, 210, 212, 213, 215, 216, 222–224, 227, 229, 231, 233, 235, 241, 243, 247, 253–255, 259, 260, 263, 266, 272, 278, 279, 282, 285, 291, 298, 299, 310, 311, 316, 317, 323, 324, 327, 330, 331, 337, 343, 344, 350, 356, 363, 364, 369, 377, 390, 398, 403, 407, 410, 424, 436, 437, 451, 452, 472, 481, 485, 486, 493, 507, 508, 514, 521, 543, 556, 557, 579, 606, 607, 629, 643, 680, 693, 694, 716, 717, 769, 807, 860, 922, 923, 1278.

Fig. 3 Indices of the 250 zeros of .s/ employed in the calculations

Using this empirical value, we could estimate the time required to obtain our desired result in Pólya’s problem. Some estimates are recorded in Table 1. For a number of choices of n, this table lists the number of zeros m required to obtain the desired result, then our estimate for the number of bits of precision b required in order to obtain a value for N near 3000, followed by the best possible result C we could obtain (that is, without the attenuating factor of N=.N C1/), and finally the estimated time required to complete this calculation, in core-days on a typical workstation. These estimates assumed ı D 0:99. Our results led us to choose n D 250. Some additional experiments indicated that we could likely trim our value of b to 6600, so our final choice of parameters was n D 250, m D 3700 (and T D 3701  ), b D 6600, and ı D 0:99. The indices ai of the n D 250 zeros selected for this computation are shown in Fig. 3. The numerical values of these zeros were computed to high precision using the mpmath Python library [15]. This package was also used to compute the values of the zeta function and its derivative that were needed for the calculation of the residues in (10). We constructed the 3451 matrices for the lattices i , and then used SageMath [28] to run each of these through LLL followed by Gram–Schmidt. After this, inequalities (16) and (17) were used to determine a lower bound Ni on the length of a nonzero vector in the lattice i . These computations were performed on two clusters, one housed in the Research Computing Group at Simon Fraser University, and another that is part of the WestGrid computing consortium of western Canada. The smallest Ni reported from these jobs was 2527; the mean value was 3720. With only 62 of these matrices producing a bound below 3100, we checked these cases a second time at a higher precision, choosing b D 6950. This raised the value for each of these cases to at least 6702, so we may use N D 3100 as our value for weak independence. We therefore obtain the following result.

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Theorem 7 The set  0 D fai W 1  i  250g, where the sequence ai is shown in Fig. 3, is 3100-independent in  \ Œ0; T, with T D 3701  1010 . Using this result in Theorem 6 with G D F0 and kT .t/ given by (18) immediately produces Theorem 1. We may now use Theorem 7 to deduce bounds on the oscillations of other sums involving the Liouville function. By choosing G D F˛ in Theorem 6 with ˛ D 1=4, 1=2, or 3=4, employing formulas for the residues shown in (9) and (10), using the kernel (18) again, and translating the results for the functions A˛ .u/ back to the sums L˛ .x/ by using (5) and (6), we obtain the following theorem. Theorem 8 Each of the following inequalities is satisfied for infinitely many integers x. L1=4 .x/ < 3:0573x1=4 ; L1=4 .x/ > 0:31824x1=4 ;

(19)

L1=2 .x/ 

log x log x < 1:1634; L1=2 .x/  > 2:2122; 2.1=2/ 2.1=2/

(20)

L3=4 .x/ 

.3=2/ .3=2/ 0:31824 3:0573 <  1=4 ; L3=4 .x/  > 1=4 : .3=4/ x .3=4/ x

(21)

It may seem that the inequalities in Theorem 8 could be improved by tuning our calculation for the specific cases ˛ D 1=2 or ˛ D 3=4. After all, the residues in (10) exhibit a dependence on ˛, so it seems possible that a different set  0 would be optimal for these choices. However, with the same choices for n and m, we find that the optimal set of indices for  0 for these cases is precisely the list shown in Fig. 3. Thus, the results of Theorem 8 cannot be improved with another calculation of the same size by using this method. The same method allows us to establish Theorem 2. Figure 4 shows the lower bound on lim supu!1 A˛ .u/, as a function of ˛, that we obtain from Theorem 6 by using the independence result of Theorem 7. This graph has a root when 1=.1=2/.12˛/ D 1:6878471 : : : , so when ˛ D 0:297148 : : : , and this establishes the first statement in the theorem. The statement that lim infu!1 A˛ .u/ is infinitely often positive for ˛ > 0:702851 : : : then follows by symmetry. As ˛ is taken to be closer to 1=2 it is harder to exhibit an infinitude of sign changes. This is not surprising since then the term Res.F˛ ; 0/ in (9) is larger and hence exerts greater dominance in (11). Indeed Humphries [13, Theorems 1.5, 1.6] has proved some conditional results about the biases in L˛ .x/ when ˛ tends to 1=2.

7 Improvements We close by briefly noting some prospects for further improvements in these oscillation bounds. As noted after the proof of Theorem 6, one could choose a bound on ci that depends on i. If, for example, jci j  N.1  i=N/, then the

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219

1.0 0.8 0.6 0.4 0.2

0.05

0.10

0.15

0.20

0.25

0.30

Fig. 4 Lower bound for lim supu!1 A˛ .u/, for 0  ˛  0:3

P sums . niD1 ci =2/2 in (15)–(17) would have a main term of .Nn/2 =16 instead of .Nn/2 =4 . This is a win; we now must balance this against the loss of attenuating the contribution from large zeros. Although these contributions are small, and hence we expect an overall win with this strategy, we have not pursued this further. We note also the qualitative gains that come from using the weak linear independence method. Prior to this method, one exhibited large oscillations in a function like L0 .x/ by searching for a single large value L0 .x0 /—cf. (4). To find an even larger oscillation, one needs to search for a larger spike in L0 .x/. There is no guarantee that such a spike will be found by extending the computational run time modestly. Indeed, given that L0 .x/ also obtains large negative values, one may have to extend the run time considerably to witness the next spike. Using this approach, it is difficult to exhibit large oscillations in L0 .x/, because L0 .x/ has large oscillations! In contrast, if one increases the size of the matrices in our method, or accounts for more zeros, then one is guaranteed a better bound in the oscillations. However, in the absence of any new idea, we believe such improvements will be small without radically larger computation times. To illustrate the marginal nature of this improvement, we note that Gonek [8] and Hejhal [12] independently conjectured P that 0 1 in terms of gamma factors. In the first, curious identity, let  denote the classical Möbius function. We point out that this is essentially a generalization of a formula for the case a D m D 1 given in Eq. (11) of [13]. Corollary 3 For all m; n  2, we have n

.n/ D m

X .k/ nk1 X k1

k

rD0

log 

1

e

 r  !! nk

m

:

The next identity gives .n/ in terms of the nth derivative of a product of gamma functions. The authors were not able to find this formula in the literature; however, given the well-known connections between  and , as well as the known example below the following theorem, it is possible that the identity is known. Theorem 2 For integers n > 1, we have .n/ D

n1 1 dn Y lim  .1  ze.j=n// : nŠ z!0C dzn jD0

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Example 1 As an example of implementing the above identity, take n D 2; then using Euler’s well-known product formula for the sine function, it is easy to check that .2/ D

1 2 d2 d2 z 1 .1 .1 lim lim D :  C z/   z/ D 2Š z!0C dz2 2Š z!0C dz2 sin.z/ 6

This last formula for .n/, following from a formula in [41] together with the preceding theorem, is analogous to some extent to the classical identity sin.n/ D ein ein . 2i Corollary 4 For integers n > 1, we have Qn1 .n/ D lim

jD0

 .1  ze.j=n// 

jD0

 .1  ze.j=n//1

2zn

z!0C

1.2.3

Qn1

:

Zeta Functions for Partitions of Fixed Length

We now consider zeta sums of the shape P .fsgk / as in Definition 2. Our first aim will be to extend (13) above (which is Corollary 2.4 of [41]). As we shall see below, these zeta values are special cases of sums considered by Hoffman in [27]. Let Œzn f represent the coefficient of zn in a power series f . Using this notation, we show the following, which in particular gives an algorithmic way to compute each P .fmgk / in terms of Riemann zeta values for m 2 N2 . Theorem 3 For all m  2, k 2 N, we have   z  1  e.r=m/  rD0 1 0 X .mj/  z mj A: D  mk Œzmk  exp @ j  j1

P .fmgk / D  mk Œzmk 

m1 Y

Generalizing the comments just below (13), the next corollary follows directly from Theorem 3 (using the well-known fact that .k/ 2 Q k for even integers k). Corollary 5 For m 2 2N even, we have that P .fmgk / 2 Q mk : Remark 2 This can also be deduced from Theorem 2.1 of [27]. To conclude this section, we note one explicit method for computing the values P .fmgk / at integral k; m (especially if m is even, in which case the zeta values below are completely elementary).

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Corollary 6 For m  2; k 2 N, and j  i, set ˛i;j WD .m.j  i C 1//

.k  i/Š :  j/Š

 m.jiC1/ .k

Then we have 0

˛1;1 ˛1;2 ˛1;3 : : : ˛1;k

B B 1 ˛2;2 B B mk  B k P .fmg / D det B 0 1 B kŠ B :: :: B : : @ 0

0

1

C ˛2;3 : : : ˛2;k C C C ˛3;3 : : : ˛3;k C C: C :: : : :: C : : C : A : : : 1 ˛k;k

Remark 3 There are results resembling these in Knopfmacher and Mays [31].

1.3 Analytic Continuation and p-Adic Continuity If we jump forward about 100 years from the pathbreaking work of Euler concerning special values of the Riemann zeta function at even integers, we arrive at the famous work of Riemann in connection with prime number theory. Namely, in 1859, Riemann brilliantly described the most significant properties of .s/ following that of an Euler product: the analytic continuation and functional equation for .s/. It is for this reason, of course, that the zeta function is named after Riemann, and not Euler, who had studied this function in some detail, and even conjectured a related functional equation. In particular, this analytic continuation allowed Riemann to bring the zeta function, and indeed the relatively new field of complex analysis, to the forefront of number theory by connecting its roots to the distribution of prime numbers. It is natural therefore, whenever one is faced with new zeta functions, to ask about their prospect for analytic continuation. Here, we offer a brief study of some of these properties, in particular showing that the situation for our zeta functions is much more singular. Partition-theoretic zeta functions in fact naturally give rise to functions with essential singularities. Here, we use Corollary 2 to study the continuation properties of partition zeta functions over partitions PmN into multiples of m > 1. In order to state the result we first define, for any " > 0, the right half-plane H" WD fz 2 C W Re.z/ > "g, and we denote by N1 the set f1=n W n 2 Ng. Corollary 7 For any " > 0 and m > 1, PmN .s/ has a meromorphic extension to H" with poles exactly at H" \ N1 . In particular, there is no analytic continuation beyond the right half-plane Re.s/ > 0, as there would be an essential singularity at s D 0.

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Remark 4 For the function PN .s/, a related discussion of poles and analytic continuation was made by the user mohammad-83 in a mathoverflow.net question. Finally, we follow Kubota and Leopoldt [33], who showed  could be modified slightly to obtain modified zeta functions for any prime p which extend  to the space of p-adic integers Zp , to obtain further examples of p-adic zeta functions of this sort. These continuations are based on the original observations of Kubota and Leopoldt, and, in a rather pleasant manner, on the evaluation formulas discussed above. In particular, we will use Corollary 6 to p-adically interpolate modified versions of P .fmgk / in the m-aspect. Given the connection discussed in Sect. 1.4 to multiple zeta values, these results should be compared with the literature on p-adic multiple zeta values (e.g., see [25]), although we note that our p-adic interpolation procedure seems to be more direct in the special case we consider. The continuation in the m-aspect of this function is also quite natural, as the case k D 1 is just that of the Riemann zeta function. Thus, it is natural to search for a suitable p-adic zeta function that specializes to the function of Kubota and Leopoldt when k D 1. It is also desirable to find a p-adic interpolation result which makes the partition-theoretic perspective clear. Here, we provide such an interpretation. Let us first denote the set of partitions with parts not divisible by p as Pp ; then we consider the length-k partition zeta values Pp .fsgk /. Note that for k D 1, Pp .fsg1 / is just the Riemann zeta function with the Euler factor at p removed (as considered by Kubota and Leopoldt). We then offer the following p-adic interpolation result. Theorem 4 Let k  1 be fixed, and let p  k C 3 be a prime. Then Pp .fsgk / can be extended to a continuous function for s 2 Zp which agrees with Pp .fsgk / on a positive proportion of integers.

1.4 Connections to Multiple Zeta Values Our final application of the circle of ideas related to partition zeta functions and infinite products will be in the theory of multiple zeta values. Definition 3 We define for natural numbers m1 ; m2 ; : : : ; mk with mk > 2 the multiple zeta value (commonly written “MZV”) .m1 ; m2 ; : : : ; mk / WD

X nm1 n1 >n2 >>nk 1 1

1 : k : : : nm k

We call k the length of the MZV. Furthermore, if m1 D m2 D    D mk are all equal to some m 2 N, we use the common notation .fmgk / WD

X n1 >n2 >>nk

1 m: .n n : 1 2 : : nk / 1

(15)

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Multiple zeta values have a rich history and enjoy widespread connections; interested readers are referred to Zagier’s short note [51], and for a more detailed treatment, the excellent lecture notes of Borwein and Zudilin [9]. There are many nice closed-form identities in the literature; for example, one can show (see [27]) on analogy to (13) that .f2gk / D

 2k : .2k C 1/Š

(16)

Similar (but more complicated) expressions for .f2t gk / for all t 2 N are given in [41], parallel to those mentioned in Sect. 1.1 for P .f2t gk /. Observe that the partition zeta function P .fmgk / (see Definition 13) can be rewritten in a similar-looking form to (15) above: X

P .fmgk / D

n1 n2 nk

1 .n1 n2 : : : nk /m 1

(17)

In fact, if we take P  to denote partitions into distinct parts, then (15) reveals .fmgk / is equal to the partition zeta function P  .fmgk / summed over length-k partitions into distinct parts, as pointed out in [41]. Series such as those in (17) have been considered and studied extensively by Hoffman (for instance, see [27]). By reorganizing sums of the shape (17), we arrive at interesting relations between P .fmgk / and families of MZVs. In order to describe these relations, we first recall that a composition is simply a finite tuple of natural numbers, and we call the sum of these integers the size of the composition. Denote the set of all compositions by C and write j j D k for D .a1 ; a2 ; : : : ; aj / 2 C if k D a1 C a2 C    C aj . Then we obtain the following. Proposition 2 Assuming the notation above, we have that P .fmgk / D

X

.a1 m; a2 m; : : : ; aj m/:

D.a1 ;:::;aj /2C j jDk

Remark 5 Proposition 2 is analogous to results of Hoffman; the reader is referred to Theorem 2.1 of [27]. In particular, for any n > 1 we can find the following reduction of .fngk / to MZVs of smaller length. We note in passing that Theorem 2.1 of [27] also shows directly how to write these values in terms of products (as opposed to simply linear combinations) of ordinary Riemann zeta values: hints, perhaps, of further connections. We remark in passing that this can be thought of as a sort of “parity result” (cf. [29, 45]). Corollary 8 For any n; k > 1, the MZV .fngk / of length k can be written as an explicit linear combination of MZVs of lengths less than k.

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As our final result, we give a simple formula for .fngk /. This formula is probably already known; if k D 2 it follows from a well-known result of Euler (see the discussion of H.n/ on p. 3 of [52]) and is closely related to (11) and (32) of [8]. The idea of the proof is also similar to what has appeared in, for example, [52]. However, the authors have decided to include it due to connections with the ideas used throughout this paper, and the simple deduction of the formula from expressions necessary for the proofs of the results described above. Proposition 3 The MZV .fngk / of length k can be expressed as a linear combination of products of ordinary  values. In particular, we have 0 1 X .nj/ k k nk nj .fng / D .1/ z exp @ z A: j j1 Remark 6 This formula is equivalent to a special case of Theorem 2.1 of [27]. However, since the proof is very simple and ties in with the other ideas in this paper, we give a proof for the reader’s convenience. The proof of Corollary 6 yields a similar determinant formula here. Corollary 9 For n  2; k 2 N, and j  i, set ˇi;j WD .n.j  i C 1//

.k  i/Š : .k  j/Š

Then we have 0

ˇ1;1 ˇ1;2 B 1 ˇ2;2 B .1/k B .fngk / D det B 0 1 B : kŠ :: @ :: : 0 0

ˇ1;3 ˇ2;3 ˇ3;3 :: :

::: ::: ::: :: :

1 ˇ1;k ˇ2;k C C ˇ3;k C C: :: C : A

: : : 1 ˇk;k

Remark 7 We can see from the above corollary that .fngk / is a linear combination of products of zeta values, which is closely related to formulas of Hoffman [27].

1.5 Machinery 1.5.1

Useful Formulas

In this section, we collect several formulas that will be key to the proofs of the theorems above. We begin with the following beautiful formula given by Chamberland and Straub in Theorem 1.1 of [12]. In fact, this formula has a long history, going back at least to Sect. 12.13 of [47], and we note that Ding, Feng, and Liu independently discovered this same result in Lemma 7 of [17].

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Theorem 5 If n 2 N and ˛1 ; : : : ; ˛n and ˇn are complex numbers, none of P ˇ1 ; : : : ;P which are non-positive integers, with njD1 ˛j D njD1 ˇj , then we have n n Y YY .k C ˛j /  .ˇj / D : .k C ˇ /  .˛j / j k0 jD1 jD1

We will also require two Taylor series expansions for log  , both of which follow easily from Euler’s product definition of the gamma function [20]. The first expansion, known as Legendre’s series, is valid for jzj < 1 (see (17) of [48]): log  .1 C z/ D  z C

X .k/ k2

k

.z/k :

(18)

We also have the following expansion valid for jzj < 2 (see (5.7.3) of [36]): log  .1 C z/ D  log.1 C z/ C z.1   / C

X

.1/k ..k/  1/

k2

zk : k

(19)

Furthermore, we need a couple of facts about Bell polynomials (see Chap. 12.3 of [2]). The nth complete Bell polynomial is the sum Bn .x1 ; : : : ; xn / WD

n X

Bn;i .x1 ; x2 ; : : : ; xniC1 /:

iD1

The ith term here is the polynomial X Bn;i .x1 ; x2 ; : : : ; xniC1 /WD

jniC1  x j1  x j2  x nŠ 1 2 niC1  ; j1 Šj2 Š    jniC1 Š 1Š 2Š .niC1/Š

where we sum over all sequences j1 ; j2 ; : : : ; jniC1 of non-negative integers such that j1 C j2 C    C jniC1 D i and j1 C 2j2 C 3j3 C    C .n  i C 1/jniC1 D n. With these notations, we use a specialization of the classical Faà di Bruno formula [22], which allows us to write the exponential of a formal power series as a power series with coefficients related to complete Bell polynomials: 0 1 1 1 X X aj j Bk .a1 ; : : : ; ak / k exp @ xAD x jŠ kŠ jD1 kD0

(20)

Faà di Bruno also gives an identity [22] that specializes to the following formula for the kth complete Bell polynomial in the series above as the determinant of a certain k  k matrix:

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0 a .k1/a .k1/a .k1/a ::: ::: a 1 1 2 3 4 k 1 2 3 k2 k2 B 1 a1 . 1 /a2 . 2 /a3 ::: ::: ak1 C C B 0 1 a1 .k3 1 /a2 ::: ::: ak2 C B Bk .a1 ; : : : ; ak / D det B 0 0 1 a1 ::: ::: ak3 C B 0 0 0 1 ::: ::: ak4 C A @ : : : : : :: : : : : : ::: :: :: :: 0

1.5.2

0

0

::: 1

0

(21)

a1

Proofs of Theorems 1 and 2, and Their Corollaries

We begin with the proof of our first main formula. Proof (Proof of Theorem 1) By (8), we find that PaCmN .n/ D

n1

Y .a C mj/n Y Y .j C 1 C a=m/n kn  : D D n n ae.r=n/ k 1 .a C mj/  1 j1 j0 rD0 j C 1 C k2aCmN Y

m

Using Theorem 5 and the well-known fact that n1 X

e.j=n/ D 0

(22)

jD0

directly gives the desired result. Proof (Proof of Corollary 1) For this, we apply (19) and use (22), the obvious fact that j.a  e.j=n//=mj < 2; and the easily checked fact that 1 C .a  e.j=n// is never a negative real number for j D 0; : : : ; n  1. Proof (Proof of Corollary 2) Here, we simply use (18). Again, the corollary is proved following a short, elementary computation, using the classical fact that n1 X rD0



e.rk=n/ D n 0

if njk; else.

Proof (Proof of Corollary 7) By Corollary 2, we find for n  2 that log .PmN .n// D

X .nk/ k1

kmkn

:

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Suppose that Re.s/ > 0 and s 62

1 . N

Then letting

K WD maxfd1= Re.s/e C 1; Re.s/g; it clearly suffices to show that X .sk/ kmks

kK

converges. But in this range on k, by choice we have Re.sk/ > 1, so that using the assumption m  2, we find for Re.s/ > 0 the upper bound X .sk/ kK

kmks

 .Ks/

X kK

1 k2k Re.s/

 .Ks/

X k1

1 k2k Re.s/

   D .Ks/ log 2 Re.s/ 2Re.s/  1 ; and note that in the argument of the logarithm in the last step, by assumption we have 2Re.s/  1 > 0. Conversely, if s 2 N1 , then it is clear that this representation shows there is a pole of the extended partition zeta function, as one of the terms gives a multiple of .1/. Proof (Proof of Corollary 3) We utilizeP a variant P of Möbius inversion, reversing the order of summation in the double sum k1 djk .d/f .nk/ks ; if g.n/ D

X f .kn/ k1

ks

;

then f .n/ D

X .k/g.kn/ k1

ks

:

Applying this inversion procedure to Corollary 2, so that g.n/ D log PmN .n/ (taking s D 1), and f .n/ D .n/=mn , we directly find that .n/ D mn

X .k/ k1

k

log .PmN .nk// :

Applying Theorem 1 then gives the result. Proof (Proof of Theorem 2) By the comments following Theorem 1.1 in [41], for M 2 N we have  Y  X zs 1 1  I 1 s D 1 C zs Q zs k s j2M 1  js k2M k2M k jk

(23)

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thus X

k

k2M

s

Q D lim

k2M

z!0C

 1 zs

 zs 1 ks

1

:

Taking M D N; s D n 2 Z2 , we apply L’Hospital’s rule n times to evaluate the limit on the right-hand side. The theorem then follows by noting, from Theorem 5, that in fact Y k2N

zn 1 n k

1 D

n1 Y

 .1  ze.j=n// :

jD0

Proof (Proof of Corollary 4) Picking up from the proof of Theorem 2 above, it follows also from Theorem 1.1 of [41] that  Y  X zs 1  s D 1  zs k

k2M

Q

j2M j 2 is prime, then  Bk1  Bk2    1  pk2 1 .mod paC1 /: 1  pk1 1 k1 k2 Let us take Ss0 to be the set of natural numbers congruent to s0 modulo p  1. The Kummer congruences then imply that for any s0 6 0 .mod p  1/, and for any k1 ; k2 2 Ss0 with k1  k2 .mod pa / and k1 ; k2 > 1, that   .1  k1 /    .1  k2 / .mod paC1 /: If we choose m1 ; m2 2 Ss0 with m1  m2 .mod pa /, then the values 1  .1  m1 /.j  i C 1/, 1  .1  m2 /.j  i C 1/ are in S1C.s0 1/.ji1/ and are congruent modulo pa , and as p > k the additional factorial terms (inside and outside the determinant) are p-integral. Now in our determinant, j  i C 1 ranges through f1; 2; : : : ; kg, and we want to find an s0 such that 1 C .s0  1/r 6 0 .mod p  1/ for r 2 f1; 2; : : : kg. If we take s0 D 2, then the largest value of 1 C .s0  1/r is k C 1, which is by assumption less than p  1, and hence not divisible by it. Hence, in our case, s0 D 2 suffices. Thus, if m1 ; m2 2 S2 with m1  m2 .mod pa /, then Pp .f1  m1 gk /  Pp .f1  m2 gk / .mod paC1 /: This shows that our zeta function is uniformly continuous on S2 in the p-adic topology. As this set is dense in Zp , we have shown that the function extends in the m-aspect to Zp .

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Proofs of Results Concerning Multiple Zeta Values

Proof (Proof of Proposition 2) Recall from (17) that we need to study the sum X n1 n2 nk

1 : .n1 n2 : : : nk /m 1

The proof is essentially combinatorial accounting, keeping track of the number of ways to split up a sum X n1 n2 nk 1

over all k-tuples of natural numbers into a chain of equalities and strict inequalities. Suppose that we have n1  n2      nk  1: Then if any of these inequalities is an equality, say nj D njC1 , in the contribution to the sum X .n1 : : : nk /m ; n1 n2 nk 1

the terms nj and njC1 “double up.” That is, we can delete the njC1 and replace the nm in the sum with an n2m . Thus, the reader will find that our goal is to keep track j j of different orderings of > and D, taking symmetries into account. The possible chains of D and > are encoded by the set of compositions of size k, by associating to the composition .a1 ; : : : ; aj / the chain of inequalities n1 D    D na1 > na1 C1 D    D na1 Ca2 > na2 C1 >    > nk : That is, the number a1 determines the number of initial terms on the right which are equal before the first inequality, a2 counts the number of equalities in the next block of inequalities, and so on. It is clear that the sum corresponding to the each composition then contributes the desired amount to the partition zeta value in the corollary. Proof (Proof of Corollary 8) In Proposition 2, comparison with Corollary 6 shows that we have a linear relation among MZVs and products of zeta values. Observe that in P .fmgk /, the only composition of length k is .1; 1; : : : ; 1/, which contributes kŠ.fmgk / to the right-hand side of Proposition 2, and that the rest of the compositions are of lower length, hence giving MZVs of smaller length; the corollary follows immediately.

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Proof (Proof of Proposition 3) Consider the multiple zeta value .fngk / of length k. Then we directly compute X

k

k

.1/ .fng /z

nk

k0

n1  z n  Y Y Y .m C 1  ze.r=n// 1 D D : m .m C 1/n m1 m0 rD0

By Theorem 5, this equals n1 Y

 .1  ze.r=n//1 :

rD0

Using precisely the same computation as was made in the proof of Theorem 3, we find that this is equal to 1

0

B X .j/ j C zC exp B A: @n j j2 njj

Hence, we have that k

.fngk / D .1/ z

nk

0 exp @

X .nj/ j1

j

1 znj A :

Proof (Proof of Corollary 9) Here we proceed exactly as in the proof of Corollary 6, except we make the simpler substitution ak D .k  1/Š.nk/ into Eq. (20), and compare with Proposition 3. In the final step, we replace the terms in the upper half of the matrix with ˇi;j as defined in the statement of the corollary.

1.6 Some Further Thoughts We have presented samples of a few varieties of flora one finds at the fertile intersection of combinatorics and analysis. What unifies all of these is the perspective that they represent instances of partition zeta functions, with proofs that fit naturally into the Eulerian theory we propound.

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We close this article by noting a general class of partition-theoretic analogs of classical Dirichlet series having the form DP 0 .f ; s/ WD

X

f . /ns ;

2P 0

where P 0 is a proper subset of P and f W P 0 ! C. Of course, partition zeta functions arise from the specialization f  1, just as in the classical case. Taking P 0 D PM as defined previously, then if f WD f .n / is completely multiplicative with appropriate growth conditions, it follows from Theorem 1.1 of [41] that we have a generalization of (8) DPM .f ; s/ D

Y j2M

f .j/ 1 s j

1 .Re.s/ > 1/ ;

(25)

and nearly the entire theory of partition zeta functions noted here (and developed in [41]) extends to these series as well.

2 Zeta Polynomials We now turn our attention toward a different layer of connections between partitions and zeta functions, via the theory of modular forms. Although Euler’s generating function for p.n/ is essentially modular, and Euler also anticipated the study of L-functions that are intimately tied to modular forms, the true depth of such observations did not come into view until further work on complex analysis was carried out in the nineteenth century. It turns out that all modular forms are related to partitions in a very direct way. Here we recall the case of this connection for modular forms for the full modular group SL2 .Z/. We then use these modular forms to define the second class of functions that are the topic of this paper, the zeta polynomials associated with modular forms.

2.1 Partitions and Modular Forms In a paper from 2004 [11], the first author, Bruinier, and Kohnen investigated the values of a certain sequence of modular functions in connection with the arithmetic properties of meromorphic modular forms on SL2 .Z/. One of the main results in the paper shows that integer partitions and a universal sequence of polynomials encode the Fourier expansions of modular forms.

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Here we recall this result, which can be thought of as a precise formulation of the assertion that a modular form is distinguished by its “first few coefficients.” In fact, by making use of partitions we have an effective recursive procedure which computes the Fourier coefficients in order. We take q WD e2iz throughout this section. Now, suppose that f .z/ D

1 X

af .n/qn

nDh

is a weight k 2 2Z meromorphic modular form on SL2 .Z/. If k  2 is even, then let Ek .z/ denote the normalized Eisenstein series Ek .z/ WD 1 

1 2k X

k1 .n/qn : Bk nD1

(26)

P Here Bk denotes the usual kth Bernoulli number and k1 .n/ WD djn dk1 . If k > 2, then Ek .z/ is a weight k modular form on SL2 .Z/; although the Eisenstein series E2 .z/ D 1  24

1 X

1 .n/qn

(27)

nD1

is not a modular form, it also plays an important role. As usual, let j.z/ denote the modular function on SL2 .Z/ which is holomorphic on H, the upper half of the complex plane, with Fourier expansion j.z/ WD q1 C 744 C 196884q C 21493760q2 C    : We will require a specific sequence of modular functions jm .z/; to define this sequence, we set j0 .z/ WD 1

and

j1 .z/ WD j.z/  744:

(28)

If m  2, then define jm .z/ by jm .z/ WD j1 .z/ j T0 .m/;

(29)

where T0 .m/ is the usual normalized mth weight zero Hecke operator. Each jm .z/ is a monic polynomial in j.z/ of degree m. Here we list the first few: j0 .z/ D 1; j1 .z/ D j.z/  744 D q1 C 196884q C    ; j2 .z/ D j.z/2  1488j.z/ C 159768 D q2 C 42987520q C    ; j3 .z/ D j.z/3  2232j.z/2 C 1069956j.z/  36866976 D q3 C 2592899910q C    :

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Let F denote the usual fundamental domain of the action of SL2 .Z/ p on H. By assumption,pF does not include the cusp at 1. Throughout, let i WD 1 and let ! WD .1 C 3/=2. If  2 F, then define e by

e WD

8 ˆ ˆ 0, .s/ D lim

N!1

23

X 1mN

ms 

N 1s 1s

At this point Riemann also describes the approximation Li.X/ to .X/ as “known”. J. E. Littlewood, Sur la distribution des nombres premiers.C. R. Acad. Sci. Paris 158 (1914), 1869–1872. 25 See W.W. Johnson, Mr James Glaisher’s factor tables and the distribution of primes, Ann. of Math. 1 (1884) 15–28. 26 See S.J. Patterson, The Riemann Hypothesis – a short history, In: Casimir Force, Casimir Operators and the Riemann Hypothesis, Eds. G. v. Dijk and M. Wakayama, de Gruyter, 2010, pp. 29–41. 27 J.L.W.V. Jensen, Sur le fonction .s/ de Riemann, Comptes rendus Acad. Sci. Paris 104 (1887) 1156–1159. In J.-P. Gram, Note sur la calcul de la fonction .s/, Oversigt over det Koniglige Danske videnskabernes Selskabs Forhandlinger og dets Medlemmers Arbejter, 1895, 303–308, the author reports that he had received in 1887 a letter from T.J. Stieltjes who had also undertaken calculations and had found the approximation 14:5 for the first zero. 28 loc. cit. 24

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combined with the Euler-Maclaurin for the same purposes. Later, in 1903, he augmented this with a summation formula due to Abel. The Finnish mathematician Ralf Josef Backlund took up the idea in his thesis of 1916 but using a variant summation formula due to Lindelöf. It seems as this work was the inspiration for Hardy and Littlewood’s “approximate function equation” of 1923. As it turned out Riemann had already found a more precise formula but which he had not published.29 What these authors were not in a position to do was to verify that all the zeros of the zeta function, in the regions examined, were on the critical line. It was only after v. Mangoldt gave a proof of the precise version of Riemann’s estimate for N.T/ in 1905 that it became possible to test the Riemann Hypothesis. He had given a weaker version in 1895 but apparently no-one attempted to develop it numerically. The tests of the Riemann Hypothesis began with the work of Backlund who developed an effective version of v. Mangoldt’s formula (1918). Finally it is of some interest to read what a well-educated young mathematician, G.B. Mathews (born in 1861) wrote about Riemann’s paper in 1892, that is, just before Hadamard’s contributions began30 : [That the formula .X/ D Li.X/ ] is approximately [i.e. empirically] correct, does not in any way prove that this is a proper asymptotic formula. It seems clear that Gauss was led to it simply by observation, and it does not appear that he ever accounted for it in any theoretical way, The only satisfactory attempt to determine a general analytical formula for F.x/ [i.e. .X/] appears to be that contained in Riemann’s celebrated memoir. This is confessedly incomplete, and the analysis which it contains is very peculiar and difficult: but because of its great importance, some account of it ought to be given. On the properties of the function  .z/ for a complex variable, which will have to be assumed in the course of the investigation, the reader may consult Prym, Zur Theorie der Gammafunction (Crelle, lxxxii. 165) and various papers by Bourguet and Mellin in vols. i, ii, iii, and viii of the Acta Mathematica.31

One can summarize these authors by saying that Riemann’s work was known and appreciated by a fair number of people in the period before the work of Hadamard but that it was not regarded as the last word, perhaps in the way that certain procedures are used in physics now which one knows work very well but where a proper deductive foundation is missing. It is also clear that before Hadamard none of these authors had gone beyond what Riemann had done. In the intervening years, however, the techniques of complex analysis had been developed and were available to a much larger group of mathematicians.

29

Carl Ludwig Siegel, über Riemanns Nachlass zur analytischen Zahlentheorie, Quellen und Studien zur Geschichte der Mathematik 2 (1932) 45–80, reprinted in [1, pp. 770–805]. Recently Wolfgang Gabcke has re-examined Riemann’s papers and reconstructed more of Riemann’s arguments. 30 Hadamard’s first paper on the zeta function was submitted for a prize offered by the Académie des Sciences de Paris in December, 1892 and published in 1893. For the history of Hadamard’s work see V. Maz’ya, T. Shaposhnikova, Jacques Hadamard, A Universal Mathematician, AMS/LMS History of Mathematics Series, 14, AMS, 1998, p. 53 ff. 31 G.B. Mathews, Theory of Numbers, Second Edition, Deighton Bell, London, 1892, Sect. 233.

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Some attempts were made to go further. Stieltjes made, in 1885, the first unjustified claim to have proved what is now called the Riemann hypothesis which shows that this was already considered to be a significant open question. What was done at this time has been absorbed into the corpus of analytic number theory and we refer to Landau’s Handbuch for the historical details.

4 Riemann’s Mode of Writing We have seen that many authors have found difficulty in unravelling some of Riemann’s arguments. There seem to be a number of reasons for this. Klein writes, rather darkly, of “Anyone who clearly understands under which Riemann worked in Göttingen. . . ”.32 Some of his difficulties are known, for example, the poverty and the illnesses plaguing his family. Klein’s formulation indicates, however, that some of the difficulties were associated with his life in Göttingen and here we have only partial information. He clearly had difficult phases but his wife, Elise, said that it was most inaccurate to represent Riemann as a hypochondriac and depressive.33 His gloomy phases, like those of Eisenstein, seem to have been also to a large extent the result of external factors about which we have relatively little information. These phases, Klein suggests, seem to have made Riemann even slower at writing up his ideas than he might have been under more favourable circumstances. Leaving this aside there is one aspect that presents itself repeatedly. This is that Riemann does not indicate what mathematics he is assuming. Thus, in his doctoral thesis, he is assuming a knowledge of topology of some sort which he does not seem to feel needs justification. There was at this time just one book on topology, Johann Benedikt Listing’s Vorstudien zur Topologie. Listing was a physicist alongside Wilhelm Weber in Göttingen. The Physikalisches Kabinett neighboured the university library and so Riemann must have encountered Listing, who, like Weber, seems to have been a gregarious person, often. While Riemann was a student Listing, a polymath of quite remarkable breadth, lectured on such topics as machines, steam engines, and meteorology. He is best known, in appropriate circles, for his work on physiological optics (he wrote the first treatise on this topic). Later he accompanied Sartorius von Waltershausen, Gauss’ first biographer, on a vulcanological expedition to carry out measurements of the magnetic field near a volcano. He gave the concept of the equipotential definition of the surface of the Earth, due to Gauss, Stokes, and others, the name “geoid”. He also introduced the word “topology” (or rather “Topologie”) into mathematics. The phrases analysis

32 33

See F. Klein, loc. cit., Preface. Katrin Scheel, Der Briefwechsel Dedekind-Weber, de Gruyter, 2014, p. 310.

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situs or geometria situs was sometimes used in the same context. This phrase had been introduced by Leibniz, but with a different meaning—see Tait’s discussion of this point.34 In his articles on Listing P.G. Tait35 has expressed the expectation that there would be much more in Listing’s papers on topology than was published in his two books but unfortunately his papers have never been edited.36 The relationship between Listing and Riemann is, unfortunately, not reflected in the papers which have been preserved and published but Listing did leave diaries in which his contacts are documented but these have not yet been made available. One can only surmise that Listing was the source of the topology Riemann assumes but even so this topology does seem to go beyond the Vorstudien.37 Listing did not lecture on topology38 (at least not officially) and we remain rather in the dark as to whether Riemann had direct contact with him in his younger years or whether he had simply studied Listing’s book carefully. The anecdote recounted by his teacher Schmalfuss39 shows how quickly Riemann could absorb and retain mathematics from a written source. Riemann’s laxity in naming his sources seems to be the best way of interpreting an incident which occurred while Riemann was preparing his Habilitationsschrift on trigonometric series. The first section gives an interesting and illuminating survey of the theory up to that point. However, Riemann records here40 that Dirichlet provided him with a list of the literature which saved him a great deal of work. Riemann was, as one can see from his works, well-read. It would seem that he could easily remember the content but forgot the source. This “help” from Dirichlet may have been a fairly heavy hint that he should mend his ways (and perhaps not adopt Gauss as a role-model). The problem in reading Riemann is then in finding out what he knew and what he was using. There are two points one should bear in mind. First of all that he belonged to a circle of physicists in Göttingen and that he was very familiar with the literature of mathematical physics. The second point is that one can also gain some information about the literature available to Riemann in Göttingen. The historical collection in the Staats- und Universitätsbibliothek in Göttingen seems to still contain what was available in Riemann’s day. On top of the literature, during

34

P.G. Tait, Scientific Papers, II, pp. 85–86. See P.G. Tait, loc. cit., Papers LXV and LXVI. 36 The editor of Johann Benedikt Listing, Beitrag zur physiologischen Optik, Ostwald’s Klassiker, 147, Leipzig 1905, Otto Schwarz has expressed similar views in the area of physiological optics, see p. 51. 37 This book influenced J. Clerk Maxwell and P.G. Tait who learnt of it from Maxwell, see P.G. Tait, loc. cit. p. 86. Its significance is also made clear in the appropriate footnotes in I. Lakatos’ Proofs and Refutations, Cambridge U.P., 1976. 38 The lecture lists of this period were published in the Göttinger Nachrichten and Listing lectured on such topics as the theory of machines, on steam engines, and on meteorology. 39 [1, p. 852; also pp. 543, 829–830]. 40 See [1, p. 578]. 35

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S.J. Patterson

his student years in Berlin, it seems to have been the lectures of Jacobi and Dirichlet that impressed Riemann most. There is no evidence of any kind of closer contact of Riemann to Jacobi but there is to Dirichlet. Since Dirichlet had over many years close connections to French mathematicians one may assume the Riemann was informed about their work even when this was not directly available to him. In the historical discussion in Riemann’s Habilitationsschrift there are many references to the French and Italian literature on the theory of Fourier series41 presumably from Dirichlet’s notes. One case where this is some interest to us is the long memoir of Binet (1839) on the gamma function. In particular Binet’s integral formula for log. .s// gives the complex version of Stirling’s formula in a very usable form.42

5 The Paper on Prime Numbers The paper on prime number can be considered as in three parts. The central idea is that there is a duality, given by an integral kernel, between the set of prime numbers and a certain set of complex numbers, the zeros of the zeta function. This is a very general theorem involving techniques applicable in other contexts. It may be seen as in the context of Riemann’s work on trigonometric series. We may see Euler’s formula sin.z/ D z

Y

.1  z2 =n2 /

m1

as the model for the introduction of the zeros of the zeta function; the ideas needed for the analytic continuation of the zeta function had been around for some time although it was Riemann who clarified the notion of “analytic continuation” in his thesis.43 With the tools he had available from complex analysis it was much easier to prove the product expansion than it had been for his predecessors, especially those of the eighteenth century. The Euler product for the zeta function, Euler’s proof of the infinitude of primes, and the theory of exponential and trigonometric functions, including the product formula for the sine function all appear in Euler’s Introductio in analysin infinitorum. The logarithmic derivative of the two sides of the product formula, a little rewritten, gives

41

Apparently Riemann did not intend to publish this work; it appeared posthumously through the offices of Dedekind a year after Riemann’s death. It has, of course, been extremely influential. 42 See E.T. Whittaker and G.N. Watson, Modern Analysis, Cambridge U.P., pp. 248–251. 43 See A. Weil, On the prehistory of the zeta-function, In: Number Theory, Trace Formulas and Discrete Groups, Symposium in Honor of Atle Selberg, Eds. K.E. Aubert, E. Bombieri, D. Goldfeld, Academic Press, 1989, 1–9.

Reading Riemann

277

 coth.x/ D

X

2x ; 2 C m2 x m2Z

a formula which is a consequence of the Poisson Summation Formula. We note, in x passing, that the easiest determination of the Fourier transform of t 7! t2 Cx 2 is by way of the calculus of residues; another is by way of the formula above. The use of special cases such as this (often in the form of kernels) to study general Fourier and trigonometric series was clear to Riemann, as his Habilitationsschrift shows. Euler’s first attempt at a proof of the product formula for the sine function was to regard the sine function as a generalized polynomial and to equate it with the product over the roots. Euler was, quite rightly, dissatisfied with this argument and in the Introductio gives a convincing proof. Riemann, like all serious analysts of his time, would have been familiar with the Introductio and it would presumably have struck him that Euler’s original argument could, in the right circumstances, be rescued. We know now the product theorems of Weierstrass and Hadamard which formalize this idea. These results show that one only needs an estimate on the growth of the function in the region considered and one can avoid the more special properties of the two sides (addition theorem, differential equation) which have been used in other proofs. From Euler’s formula one can develop the theory of the Bernoulli numbers and polynomials. One has with these tools both what one needs to develop the EulerMaclaurin summation formula and the Fourier expansion of the periodic extension of the Bernoulli polynomials from Œ0; 1 to R. One approach to this is due to Poisson (1823).44 The second part of Riemann’s paper, which is a mere sketch, is concerned with the problem of determining the zeros of more precisely. He gives the asymptotic formula and indicates that it is possible to compute the zeros. He even goes further and suggests that “the majority” of the zeros lie on the critical line. At least the second of the two integral representations of the zeta function given in the paper is clearly part of this programme. We know from the Nachlass that Riemann did do some calculations.45 There is an oral tradition of a “black notebook” of Riemann’s

44

Poisson’s argument, where he derives the Euler-Maclaurin formula from the Fourier expansion, is summarized by G.H. Hardy in his book Divergent Series, Oxford U.P. 1949, Sect. 13.8. Poisson’s original work is “Sur le calcul numérique des Intégrales définies”, Mémoires de L’Académie Royale des Sciences de l’Institut de France, Année 1823, T. VI, published 1827, 571–602. It is worth remarking that Riemann ignored some of Poisson’s real achievements on the theory of Fourier series in the introduction to his Habilitationsschrift. Poisson introduces the Poisson kernel, already an innovation, and uses it to demonstrate that for a continuous function the Fourier series is what Hardy calls summation help of Abel’s theorem (1826) P (A) to the function. Alone, and P with theP which states that if an converges then limx!1;x1

where .s/ is the Riemann zeta function and c .s/ is the imprimitive Dedekind zeta function—imprimitive because the Euler factors at the primes dividing c are removed—of the cyclotomic field Kc generated over Q by the cth roots of unity. Thus 1 .s/ is .s/ itself, 2 .s/ is .1  2s /.s/, and so on. The infinite product converges for 2. In this note we modify (1) slightly so as to obtain a product for .s/ rather than .s  1/=.s/. Let KcC be the maximal totally real subfield of Kc , and let cC .s/ be the Dedekind zeta function of KcC with the Euler factors at the primes dividing c removed. Then Y cC .s C 1/: (2) .s/ D c>1

D.E. Rohrlich () Department of Mathematics and Statistics, Boston University, Boston, MA 02215, USA e-mail: [email protected] © Springer International Publishing AG 2017 H. Montgomery et al. (eds.), Exploring the Riemann Zeta Function, DOI 10.1007/978-3-319-59969-4_12

287

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D.E. Rohrlich

Like the traditional Euler product .s/ D

Y

.1  ps /1 ;

(3)

p

the Taniyama product (2) converges for 1. The main result of this note can be viewed as an analogue of Mertens’ theorem [2]. It bears the same relation to (2) as Mertens’ theorem does to (3), and Mertens’ theorem itself figures prominently in the proof. Let  denote the Euler–Mascheroni constant. Q C  Theorem 1 c6x c .2/  e log x. Of course cC .2/ can be computed explicitly in terms of generalized Bernoulli numbers. For a primitive Dirichlet character of conductor q, let b2; D

q X

.j/.j2 =q  j C q=6/:

jD1

Also write dcC for the discriminant of KcC , and let '.c/ be the cardinality of .Z=cZ/ . For the sake of a succinct formula we put ( .c/ D

'.c/

if c > 3

2

if c D 1 or 2:

Then cC .2/ D

  .c/ Y Y

dcC 3=2

qjc mod q

.1/D1

b2;

Y

.1  .p/p2 /;

(4)

pjc

where the asterisk indicates that in the second product, runs over primitive characters of conductor q. Expressions similar to (4) have arisen in other contexts. For example, nearly the same triple product occurs in a formula of Yu [6] for the order of a certain cuspidal divisor class group of the modular curve X1 .N/ (see also Yang [5, p. 521]). Even so, the differences between Yu’s formula and (4) appear to be significant enough to preclude a straightforward interpretation of Theorem 1 as an asymptotic average of cuspidal divisor class numbers. It is a pleasure to thank the referee for a careful reading of the manuscript.

Taniyama Product

289

2 Taniyama’s Identity A proof of (1) is of course subsumed in Taniyama’s proof of his general formula, but we will nonetheless sketch a proof here before going on to the modification (2). For a prime p not dividing c let f .p; c/ be the order of the residue class of p in .Z=cZ/ . Also, write Z.s/ for the right-hand side of (1). Then Z.s/ can be written as the double product Z.s/ D

YY .1  psf .p;c/ /'.c/=f .p;c/ ;

(5)

c>1 p−c

where the inner product is c .s/ and runs over primes not dividing c. The proof of (1) amounts to reversing the order of multiplication in the double product in (5). By choosing a branch of log Z.s/ we can do the computation additively, and the absolute convergence of the resulting triple sum in the right half-plane 2 will show a posteriori that the original double product is meaningful in this region and that the calculation is legitimate. We define our branch of log Z.s/ by log Z.s/ D

X X '.c/ X pmf .p;c/s : f .p; c/ m>1 m c>1 p−c

Putting d D mf .p; c/ and summing over d > 1, we obtain log Z.s/ D

X X X '.c/ pds d c>1 d>1 p−c f .p;c/jd

or equivalently (since f .p; c/jd if and only if pd  1 mod c) log Z.s/ D

X X X '.c/ pds : d p d>1 c>1 cjpd 1

As

P

mjn

'.m/ D n, we conclude that log Z.s/ D

XX p

d1 pds .pd  1/:

d>1

The inner sum equals log..1  p1s /1 .1  ps //, and (1) follows.

(6)

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D.E. Rohrlich

3 The Modification The proof of (2) is much the same. Put ' C .c/ D ŒKcC W Q, so that ' C .c/ is 1 if c D 1 or 2 and '.c/=2 otherwise. (Note also that ' C .c/ D .c/=2, where .c/ is as in the introduction.) If p is a prime not dividing c, then the order in Gal.KcC =Q/ of a Frobenius at p will be denoted f C .p; c/. Of course f C .p; c/ is also the order of the coset represented by p in the quotient of .Z=cZ/ by the image of f˙1g. Let Z C .s/ denote the right-hand side of (2), and write Z C .s/ as a double product: YY C C C Z C .s/ D .1  psf .p;c/ /' .c/=f .p;c/ : (7) c>1 p−c

Define a branch of log Z C .s/ by setting log Z C .s/ D

X X ' C .c/ X pmf C .p;c/s : f C .p; c/ m>1 m c>1

(8)

p−c

The calculation will again show that the triple sum is absolutely convergent for 2. Putting d D mf C .p; c/ and summing over d > 1, we find log Z C .s/ D

XX X c>1 d>1

p−c f C .p;c/jd

' C .c/ ds p d

as before. But the condition f C .p; c/jd means cjpd  1 or cjpd C 1, so we get log Z C .s/ D

X X X ' C .c/ pds : d d p d>1

(9)

cj.p ˙1/

We emphasize that the innermost sum is the sum over all c such that at least one of the conditions cjpd  1 and cjpd C 1 is satisfied. If c > 3, then the conditions cjpd  1 and cjpd C 1 are mutually exclusive and C ' .c/ D '.c/=2, so we have X

' C .c/ D

c>3 cjpd ˙1

1 X 1 X '.c/ C '.c/: 2 c>3 2 c>3 cjpd 1

(10)

cjpd C1

On the other hand, if c D 1 or 2, then the conditions cjpd  1 and cjpd C 1 are both satisfied (for if c D 2, then p is odd), P but ' C .c/ D 1. So Eq. (10) is correct without the restriction c > 3, and the identity mjn '.m/ D n gives X cj.pd ˙1/

' C .c/ D

1 d ..p  1/ C .pd C 1// D pd : 2

(11)

Taniyama Product

291

Multiplying through by pds =d in (11) and inserting the result in (9), we obtain log Z C .s/ D

X

log.1  p1s /1 ;

p

or in other words Y

cC .s/ D .s  1/

(12)

c>1

for 2. Replacing s by s C 1 gives (2) for 1.

4 The Analogue of Mertens’ Theorem We shall prove that Y

cC .2/  e log x:

(13)

c6xC1

Theorem 1 is an immediate consequence of (13), because log.x  1/  log x. We proceed as in the derivation of (2), but with two crucial changes: first, we take s D 2, and second, c now runs over the finite set of positive integers 6 x C 1. Thus (8) is replaced by log

Y

cC .2/ D

c6xC1

X X ' C .c/ X p2mf C .p;c/ : f C .p; c/ m>1 m c6xC1

(14)

p−c

Next we make the change of variables d D mf C .p; c/. Since c runs over a finite set and the Dirichlet series for log cC .s/ is absolutely convergent for 1 and in particular for s D 2, we can rearrange the order of summation to obtain log

Y

cC .2/ D

c6xC1

XX X p

' C .c/

d>1 c6xC1 cjpd ˙1

p2d d

(15)

as in (9). For the sake of notational simplicity, we conflate the double sum over p and d into a single sum over pd , and we put ˚.pd ; x/ D

p2d X C ' .c/: d c6xC1 cjpd ˙1

(16)

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D.E. Rohrlich

Then (15) can be written in the form Y X X X log cC .2/ D ˚.pd ; x/ C ˚.p; x/ C ˚.pd ; x/: p>x

pd >x d>2

c6xC1

(17)

pd 6x

We shall prove the following assertions: X ˚.pd ; x/ D o.1/:

(18)

pd >x d>2

X

˚.p; x/ D o.1/:

(19)

p>x

X

˚.pd ; x/ D log

Y .1  p1 /1 C o.1/:

(20)

p6x

pd 6x

Granting these statements and using them in (17), we find that Y Y log cC .2/ D log .1  p1 /1 C o.1/; p6x

c6xC1

whence exponentiation and an appeal to Mertens’ theorem give (13). To prove (18), we first note that X ' C .c/ 6 pd :

(21)

c6xC1 cjpd ˙1

by (11). Thus ˚.pd ; x/ 6 pd =d by (16), whence the left-hand side of (18) is d bounded P P bydthe sum of the terms with p > x in the convergent double series p d>2 p =d: Since the tail of a convergent series is o.1/, we obtain (18). Next we prove (19). Take x > 20. It suffices to show that the sums X X D ˚.p; x/ 1

xx log x

are both o.1/. Appealing once again to (21) and (16), we see that X X 6 p1 D log.log.x log x/=.log x// C o.1/ 1

x 20 we have .x log x/=c > e; hence the integrand is 6y2 and the integral is 6.x log x/1 . It follows that the sum over c is 6.1 C 1=x/=.log x/ and thus o.1/. Finally we prove (20). The summation on the left-hand side of (20) is restricted to pd 6 x, so if cjpd ˙ 1, then c 6 x C 1. Hence ˚.pd ; x/ coincides with pd =d by (11) and (16), so X

˚.pd ; x/ D

pd 6x

X

pd =d:

pd 6x

We may write this identity as X pd 6x

˚.pd ; x/ D

X X p6x d6 log x log p

pd =d;

(25)

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D.E. Rohrlich

while log

Y

.1  p1 /1 D

p6x

XX

pd =d:

(26)

p6x d>1

Subtracting (25) from (26), we see that log

Y

.1  p1 /1 

p6x

X

˚.pd ; x/ D

XX p6x

pd 6x

pd =d

(27)

pd >x

If p 6 x and pd > x, then d > 2, so (27) gives log

Y X X .1  p1 /1  ˚.pd ; x/ 6 pd =d: p6x

pd 6x

pd >x d>2

The left-hand side is positive by (27), and as noted previously, the right-hand side is the tail of a convergent double series, and therefore o.1/. Hence the left-hand side is o.1/, and (20) follows.

5 The Special Value For the sake of completeness, we recall the standard calculation of cC .2/ in terms of generalized Bernoulli numbers. Write cC .s/ as a product of Dirichlet L-functions associated with even Dirichlet characters to the modulus c: Y L.s; /: (28) cC .s/ D

mod c

.1/D1

We restrict attention to primitive characters by writing cC .s/ D

 Y Y qjc mod q

.1/D1

L.s; /

Y

.1  .p/ps /:

(29)

pjc

Now recall the functional equation of L.s; /: For even and primitive of conductor q, let .s; / D qs=2  s=2  .s=2/L.s; /I

(30)

.s; / D W. /.1  s; /;

(31)

then

Taniyama Product

295

where W. / is the root number of . On the other hand, according to a classic formula we have L.1  k; / D bk; =k for integers k > 2 (and actually even for k D 1 if ¤ 1). Taking k D 2 and applying (31), we obtain L.2; / D  2 b2; W. /=q3=2 :

(32)

Next recall that if has order > 3 then W. /W. / D 1, while if 2 D 1, then W. / D 1. Thus on substituting (32) in (29), we obtain cC .2/ D

Y

. 2 =q3=2 /

 Y

C .q/

b2;

mod q

.1/D1

qjc

Y

.1  .p/p2 /;

(33)

pjc

where C .q/ is the Pnumber of even Dirichlet characters which are primitive of conductor q. Since qjc C .q/ D ' C .c/ we have Y

. 2 /

C .q/

D  .c/ :

(34)

D dcC ;

(35)

qjc

Furthermore Y

q

C .q/

qjc

as one sees, for example, by observing that the exponential factor in the functional equation of the Dedekind zeta function of KcC is .dcC /s=2 , while the exponential factor in (31) or rather (30) is qs=2 . On substituting (34) and (35) in (33), we obtain (4).

6 A Question For c > 3, let c be the quadratic Hecke character of KcC associated with the extension Kc =KcC , and let L.s; c / be the corresponding Hecke L-function. Write Lc .s/ for the imprimitive Hecke L-function obtained from L.s; c / by deleting the Euler factors at the primes dividing c. Also put Lc .s/ D 1 for c D 1 or 2. Then Lc .s/ D c .s/=cC .s/ in all cases. Hence combining (1) with (12), we obtain .s/1 D

Y

Lc .s/:

c>1

The infinite product converges for 2, but 1=.s/ is holomorphic and nonvanishing for 1. Is the true region of convergence perhaps much larger than 2? We can offer only a minimal enlargement:

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D.E. Rohrlich

Theorem 2 The product

Q

c>1

Lc .s/ converges to .s/1 for 2.

Proof For integers c > 3 and primes p − c, put .p; c/ D c .p/, where p is a prime ideal of KcC lying above p. If c D 1 or 2 put .p; c/ D 0. Then Lc .s/ D

Y

.1  .p; c/pf

C .p;c/s

/'

C .p;c/=f C .p;c/

p−c

for 1, and consequently Y

log

Lc .s/ D

c6xC1

mf C .p;c/s X X ' C .c/ X m p : .p; c/ f C .p; c/ m>1 m c6xC1 p−c

Making the change of variables d D mf C .p; c/ as before, we obtain Y

log

LcC .s/ D

XX X p

c6xC1

' C .c/ .p; c/d=f

C .p;c/

d>1 c6xC1 cjpd ˙1

pds d

(note that the condition cjpd ˙ 1 means precisely that f C .p; c/jd). All of this is valid for 1, but we now assume that 2 or simply that 3. The C conditions pd  1 mod c and .p; c/d=f .p;c/ D 1 are equivalent, because both are equivalent to the assertion that f .p; c/ D 2f C .p; c/ and d=f C .p; c/ is odd. It follows that the complementary conditions, namely pd  1 mod c and C .p; c/d=f .p;c/ D 1, are also equivalent, whence X

' C .c/ .p; c/d=f

C .p;c/

D

c>3 cjpd ˙1

1 X 1 X '.c/  '.c/: 2 c>3 2 c>3 cjpd 1

(40)

cjpd C1

As before, the restriction c > 3 can be eliminated throughout (40), because if c D 1 or 2, then .p; c/ D 0. With the restriction c > 3 removed, (40) implies that X

' C .c/ .p; c/d=f

cjpd ˙1

C .p;c/

D

1 d ..p  1/  .pd C 1// D 1; 2

and therefore (39) gives X

 .pd ; x; s/ D 

pd 6x

X X pds : d log x p6x

(41)

d6 log p

On the other hand, log .s/1 D 

X X pds p

d>1

d

(42)

Taking the absolute value of the difference of (41) and (42), we find ˇX ˇ X ds ˇ ˇ p d 1 ˇ ˇ :  .p ; x; s/  .s/ ˇ 6 ˇ d d d p 6x

p >x

Since 2 the right-hand side is the tail of a convergent series (namely the Dirichlet series for log .s/) and is therefore o.1/. Thus (38) follows and the proof of the theorem is complete.

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D.E. Rohrlich

References 1. P. Chebyshev, Sur la fonction qui détermine la totalité des nombres premiers inférieurs à une limite donnée. Mém. Acad. Imp. Sci. Pétersbourg 6, 141–157 (1851) 2. F. Mertens, Ein Beitrag zur analytischen Zahlentheorie. J. Reine Angew. Math. 78, 46–62 (1874) 3. H.L. Montgomery, R.C. Vaughan, The large sieve. Mathematika 20, 119–134 (1973) 4. Y. Taniyama, L-functions of number fields and zeta functions of Abelian varieties. J. Math. Soc. Jpn. 9, 330–366 (1957) 5. Y. Yang, Modular units and cuspidal divisor class groups of X1 .N/. J. Algebra 322, 514–553 (2009) 6. J. Yu, A cuspidal class number formula for the modular curves X1 .N/. Math. Ann. 252, 197–216 (1980)